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STUDY GUIDE

GRADE 10 – 1ST SEM | UNIT 7

Polynomial Equations

Table of Contents

Introduction ...................................................................................................................................... 3

Test Your Prerequisite Skills .......................................................................................................... 4

Objectives ........................................................................................................................................ 5

Lesson 1: Translating Real-life Situations into Polynomial Equations

- Warm Up! ............................................................................................................................. 6

- Learn about It! ..................................................................................................................... 7

- Let’s Practice! ....................................................................................................................... 7

- Check Your Understanding! ............................................................................................. 14

Lesson 2: The Rational Root Theorem

- Warm Up! ........................................................................................................................... 15

- Learn about It! ................................................................................................................... 16

- Let’s Practice! ..................................................................................................................... 18


Lesson 3: Solving Polynomial Equations

- Warm Up! ........................................................................................................................... 24

- Learn about It! ................................................................................................................... 25

- Let’s Practice! ..................................................................................................................... 28


Lesson 4: Solving Problems Involving Polynomial Equations

- Warm Up! ........................................................................................................................... 38

- Learn about It! ................................................................................................................... 39

- Let’s Practice! ..................................................................................................................... 39



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Challenge Yourself! ....................................................................................................................... 46

Performance Task ......................................................................................................................... 47

Wrap-up ......................................................................................................................................... 48

Key to Let’s Practice! ...................................................................................................................... 50

References ..................................................................................................................................... 51


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STUDY GUIDE

GRADE 10 – 1ST SEM|MATHEMATICS

UNIT 7

Polynomial Equations Have you ever been on a roller coaster? These rides are exciting because they run through curves, peaks, and dips at high speeds. The designers of roller coasters sometimes build roller coaster tracks based on the graphs of polynomials. For example, the graph on the right resembles a path of a roller coaster. This is modeled by the function 𝑅(𝑥) = 0.001𝑥) − 0.24𝑥- + 4𝑥 + 12.

Polynomials are also used by economists. For instance, combinations of polynomial functions are used to do cost analyses. Moreover, polynomials can be used to model stock market situations to see or predict how stocks vary over time. People in business also use polynomials to see how will

their sales be affected if the price of their goods will be raised. Healthcare professionals also require skills in polynomial equations. They use polynomials to keep record of a patient’s progress and to further determine their schedule. For instance, the weight 𝑊 of a sick patient can be modeled by the equation 𝑊(𝑛) = 0.1𝑛) − 0.5𝑛- + 121 where 𝑛 is the number of weeks since the said patient became ill. This of course varies from one person to another and only models the person’s recorded weight. These are just some of the exemplifications that can be modeled by polynomials. In this unit, you will learn more about polynomials and some of its representations.

Click Home icon to go back to Table of Contents


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STUDY GUIDE

Before you get started, answer the following items on a separate sheet of paper. This will help you assess your prior knowledge and practice some skills that you will need in studying the lessons in this unit. Show your complete solution.

1. Translate the following sentences to mathematical sentences. a. Eight less than a number is 5. b. The sum of two consecutive numbers is 19. c. Two less than thrice a number is 10. d. The length of a rectangle is twice its width. e. The radius of a cylinder is 5 less than the measurement of the cylinder’s

height.

2. Which of the following expressions is a factor of the polynomial function 𝑃(𝑥) = 𝑥3 + 2𝑥4 − 10𝑥) − 20𝑥- + 9𝑥 + 18?

a. (𝑥 + 1) b. (𝑥 − 2) c. (𝑥 − 3) d. (𝑥 + 6) e. (𝑥 − 9)

Test Your Prerequisite Skills

• Translating English phrases/sentences to mathematical phrases/sentences

• Identifying whether a given expression is a factor of a polynomial using the Factor Theorem

• Dividing polynomials using synthetic division • Factoring different types of polynomials


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STUDY GUIDE

3. Divide the following polynomials using synthetic division. a. (𝑥- + 2𝑥 − 35) ÷ (𝑥 − 5) b. (3𝑥- − 𝑥 − 10) ÷ (𝑥 − 2) c. (𝑥) − 1) ÷ (𝑥 − 1) d. (𝑥) + 5𝑥- − 4𝑥 − 20) ÷ (𝑥 + 5) e. (𝑥) + 6𝑥- + 9𝑥 + 4) ÷ (𝑥 + 4)

4. Factor the following polynomials.

a. 6𝑥) − 3𝑥- b. 81𝑥- − 49𝑦- c. 8𝑥) + 27𝑦) d. 2𝑥- − 16𝑥 + 32 e. 9𝑥- + 15𝑥 + 4

At the end of this unit, you should be able to

• translate real-life situations into polynomial equations; • find the rational roots of a polynomial equation; • find the polynomial equation given its roots; • solve polynomial equations; and • solve problems involving polynomial equations.

Objectives


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STUDY GUIDE

Four-Window Foldable Materials Needed: colored papers and pens Instructions:

1. This activity may be done individually or in pairs. 2. Fold the colored paper lengthwise, but allot a 1-inch from the edge of the

paper. 3. Create a 4-window foldable as shown below.

4. List at least five words or phrases associated to the given operation in each window.

5. Design your foldable afterwards.

Lesson 1: Translating Real-life Situations into Polynomial Equations

Warm Up!

CLUE WORDS/PHRASES

Addition Subtraction Multiplication Division


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STUDY GUIDE

Many real-life objects and phenomena can be represented by polynomials. The words you listed in the Warm Up! activity will help you translate some practical situations into polynomial equations.

If a polynomial involves the variable 𝑥, it can be written in the form

𝑎=𝑥= + 𝑎=>?𝑥=>? + ⋯+ 𝑎?𝑥 + 𝑎A

where 𝑎A, 𝑎?,⋯ , 𝑎= are real numbers such that 𝑎= ≠ 0 and 𝑛 is a nonnegative integer.

Example 1: Write the polynomial that represents the volume (in cubic meters) of the

wooden block in the given figure in terms of its height 𝑥.

Let’s Practice!

Learn about It!

Definition 1.1: A polynomial is an expression that contains variables and constants which can be combined using addition, subtraction, and multiplication.


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STUDY GUIDE

Solution:

Step 1: Identify the working formula. The wooden block is in the shape of a rectangular prism whose volume 𝑉 is

given by the formula

𝑉 = 𝑙𝑤ℎ where 𝑙 is the length of the rectangular prism, 𝑤 is its width, and ℎ is its

height.

Step 2: List the relevant expressions. According to the figure, the expressions that represent the length, width, and

height of the block are 𝑙 = 13𝑥 − 11, 𝑤 = 13𝑥 − 15, and ℎ = 𝑥, respectively.

Step 3: Set up the algebraic model as a polynomial equation. Substituting the given dimensions into the formula and multiplying the

expressions results in the following solution:

𝑉 = 𝑙𝑤ℎ 𝑉 = (13𝑥 − 11)(13𝑥 − 15)(𝑥) 𝑉 = 169𝑥) − 338𝑥- + 165𝑥

Therefore, the volume of the wooden block is represented by

𝑉 = (169𝑥) − 338𝑥- + 165𝑥)cubic meters.


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STUDY GUIDE

Try It Yourself! Write the polynomial that represents the area of the cardboard in the figure in

terms of its length 𝑥. Example 2: The radius of a cylinder is given by the expression (2𝑥 − 1) meters and its

height is (4𝑥- − 25) meters. Construct a polynomial equation that will best describe its surface area.

Solution:

Step 1: Identify the working formula. The formula for the surface area 𝑆𝐴 of a cylinder is given by

𝑆𝐴 = 2𝜋𝑟- + 2𝜋𝑟ℎ where 𝑟 is the radius of the circle and ℎ is the height of the cylinder.

Step 2: List the relevant expressions. Let 𝑟 = 2𝑥 − 1 and ℎ = 4𝑥- − 25.


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STUDY GUIDE

Step 3: Set up the algebraic model as a polynomial equation. Substitute the given dimensions into the formula, and simplify.

𝑆𝐴 = 2𝜋𝑟- + 2𝜋𝑟ℎ 𝑆𝐴 = 2𝜋(2𝑥 − 1)- + 2𝜋(2𝑥 − 1)(4𝑥- − 25) 𝑆𝐴 = 2𝜋(4𝑥- − 4𝑥 + 1) + 2𝜋(8𝑥) − 4𝑥- − 50𝑥 + 25) 𝑆𝐴 = 8𝜋𝑥- − 8𝜋𝑥 + 2𝜋 + 16𝜋𝑥) − 8𝜋𝑥- − 100𝜋𝑥 + 50𝜋 𝑆𝐴 = 16𝜋𝑥) − 108𝜋𝑥 + 52𝜋

Therefore, the surface area of the cylindrical tank is represented by the

equation 𝑆𝐴 = (16𝜋𝑥) − 108𝜋𝑥 + 52𝜋) square meters. Try It Yourself! The length of a rectangle is twice its width. Write the polynomial that represents the

area of the rectangle in terms of its width 𝑥. Example 3: The edge of the base of a square pyramid is 5 ft shorter than its height.

Represent the volume of the pyramid in terms of its height 𝑥. Solution:

Step 1: Identify the working formula. The formula for the volume 𝑉 of a square pyramid is given by

𝑉 =13𝑏

-ℎ

where 𝑏 is the length of one edge of the base of the pyramid and ℎ is its height.


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STUDY GUIDE

Step 2: List the relevant expressions. If the edge of the pyramid’s square base is 5 ft shorter than its height 𝑥, then

𝑏 = 𝑥 − 5.

Step 3: Set up the algebraic model as a polynomial equation by substituting the given dimensions into the formula, then simplifying the expressions.

𝑉 =13𝑏

-ℎ

𝑉 =13(𝑥 − 5)-(𝑥)

𝑉 =13(𝑥- − 10𝑥 + 25)(𝑥)

𝑉 =13𝑥

) −103 𝑥- +

253 𝑥

The volume of the square pyramid is represented by

𝑉 = N?)𝑥) − ?A

)𝑥- + -3

)𝑥Ocubic feet.

Try It Yourself!

The base of a triangular pyramid is an equilateral triangle. If the height of the pyramid is thrice the length of the base 𝑥, then what is the polynomial equation that will represent the volume of the pyramid?


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STUDY GUIDE

Real-World Problems Example 4: A box with an open top is to be made from a

rectangular sheet of cardboard that measures 25 inches by 20 inches. Squares from each corner will be cut, and the sheets will then be folded and secured to form the sides of the box. What expression will represent the volume of the box in terms of the length 𝑥 of the square?

Solution:

Step 1: Identify the working formula. The box is in the shape of a rectangular prism whose volume 𝑉 is given by

the formula


height.

Step 2: List the relevant expressions. If the length of the square to be cut from each corner of the sheet is 𝑥, then

the dimensions of the box would be 𝑥, 25 − 2𝑥, and 20 − 2𝑥 as shown below:


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STUDY GUIDE

Step 3: Set up the algebraic model as a polynomial equation. When we substitute the expressions for each dimension, we obtain the

following solution:

𝑉 = 𝑙𝑤ℎ 𝑉 = (25 − 2𝑥)(20 − 2𝑥)(𝑥) 𝑉 = 4𝑥) − 90𝑥- + 500𝑥

Therefore, the volume of the box is represented by𝑉 = (4𝑥) − 90𝑥- + 500𝑥)

cubic inches.

Try It Yourself!

A square-sized wooden tray is to be made by cutting four small squares with side 𝑥 from its corners that are equal in size. This will then be folded up the sides, and taped in the corners. If the side of the cardboard measures 8 inches, what expression will represent the volume of the tray?


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STUDY GUIDE

1. Write the polynomial that represents the surface area of the wooden block in the figure in terms of its height 𝑥.

2. The length of a rectangle is twice its width. Write the polynomial that represents the

perimeter of the rectangle in terms of its width 𝑥.

3. A cylindrical tank was bought to be used for water storage. The radius of the tank is (2x − 1) meters and its height is (4𝑥- − 25) meters. Construct a polynomial equation that will describe its volume.

4. The width of a storage compartment is 8 ft more than its height, and its length is 5 ft less than its height. If the volume of the compartment is 120 ft3, write its polynomial equation in standard form.

5. A museum intends to put one of its artifacts on display in a pyramid-shaped glass case. The height of the case is thrice as tall as the artifact and the edge of its square base is 5 ft shorter than the artifact’s height. Represent the volume of the pyramid in terms of the artifact’s height 𝑥.

Check Your Understanding!


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STUDY GUIDE

Observing Factors and Zeros Materials Needed: activity sheet, pen, paper Instructions:

1. This activity may be done individually or in pairs. 2. You will be given two polynomials in which you are to find its factors.

𝑷(𝒙) in general form 𝑃(𝑥) = 𝑥- − 1 𝑃(𝑥) = 𝑥- + 7𝑥 + 12 𝑷(𝒙) in factored form

3. Equate each of the factors to 0. These are called the roots of the equation.

𝑷(𝒙) in general form 𝑃(𝑥) = 𝑥- − 1 𝑃(𝑥) = 𝑥- + 7𝑥 + 12 𝑷(𝒙) in factored form

Roots

4. Let 𝑎A be the constant term and 𝑎= be the leading coefficient of the polynomial 𝑃(𝑥). Identify the factors of 𝑎A and 𝑎=. Factors of 𝑎A: __________ Factors of 𝑎=: __________

5. Let 𝑝 be the factors of 𝑎A and 𝑞 be the factors of 𝑎=. List the possible values of UV

by forming all possible combinations of 𝑝 and 𝑞. 𝑝𝑞 = _____

6. Compare the roots of the equation and the possible values of UV. What have you

observed?

Lesson 2: The Rational Root Theorem

Warm Up!


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STUDY GUIDE

The activity you did in Warm Up! states the Rational Root Theorem. This theorem limits the possible rational roots that you may test on a given polynomial equation. Note that if a number 𝑛 is a solution of 𝑃(𝑥) = 0, then 𝑛 is called a root of the equation. Additionally, if a number 𝑛 makes a polynomial function 𝑃(𝑥) zero when it is substituted for 𝑥, then 𝑛 is called a zero of the function. A polynomial equation is said to be in general form if it is written as

𝑎=𝑥= + 𝑎=>?𝑥=>? + ⋯+ 𝑎?𝑥 + 𝑎A = 0 where 𝑎A, 𝑎?,⋯ , 𝑎= are real numbers such that 𝑎= ≠ 0 and 𝑛 is a nonnegative integer. Once a zero is obtained, we can use the Factor Theorem to identify the factors of 𝑃(𝑥).

Learn about It!

Rational Root Theorem Suppose 𝑃(𝑥) is a polynomial function whose leading term is 𝑎=𝑥= and whose constant term is 𝑎A, where 𝑎= and 𝑎A are nonzero integers. If the rational number U

V is a zero

of 𝑃(𝑥) such that the GCF of 𝑝 and 𝑞 is 1, then 𝑝 is a factor of 𝑎A, and 𝑞 is a factor of 𝑎=.

Factor Theorem If 𝑃(𝑥) is a polynomial function and 𝑘 is a real number, 𝑥 − 𝑘 is a factor of 𝑃(𝑥) if and only if 𝑃(𝑘) = 0.


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STUDY GUIDE

Moreover, the Fundamental Theorem of Algebra states that a polynomial function 𝑃(𝑥) of degree 𝑛 has exactly 𝑛 complex zeros. This theorem will help you check if you have the correct number of zeros. Example: What are the possible rational roots of the polynomial equation 𝑓(𝑥) = 0

where𝑓(𝑥) = 𝑥) − 2𝑥- + 4𝑥 − 8? Solution:

Step 1: Arrange the terms in decreasing order of degree. This step is important so that you have the correct values for 𝑎A and 𝑎=. This is already the case in the given example, so proceed to the next step.

Step 2: Determine the values of 𝑎A and 𝑎=. In this case, 𝑎A = −8 and 𝑎= = 1. Step 3: List the factors of 𝑎A and 𝑎= to obtain the possible values of 𝑝 and 𝑞,

respectively. Be sure to consider both positive and negative values. 𝑝 = ±1,±2,±4,±8

𝑞 = ±1

Step 4: List the possible values of UV by forming all possible combinations of the

values of 𝑝 and 𝑞. Again, consider both positive and negative values. You may eliminate duplicates as well.

𝑝𝑞 = ±1,±2,±4,±8

Therefore, the possible rational roots of the polynomial equation 𝑓(𝑥) = 0

where 𝑓(𝑥) = 𝑥) − 2𝑥- + 4𝑥 − 8 are ±1,±2,±4,±8.


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STUDY GUIDE

Example 1: List all the possible rational zeros of 𝑓(𝑥) = 2𝑥) − 7𝑥- − 17𝑥 + 10. Solution: Recall that if a number 𝑛 makes a polynomial function 𝑓(𝑥) zero when it is

substituted for 𝑥, then 𝑛 is called a zero of the function. We can use the Rational Root Theorem to determine the possible rational roots of the polynomial equation 𝑓(𝑥) = 0.

Step 1: Arrange the terms in decreasing order of degree. This is already the case in

the given example, so proceed to the next step. Step 2: Determine the values of 𝑎A and 𝑎=. In this case, 𝑎A = 10 and 𝑎= = 2. Step 3: List the factors of 𝑎A and 𝑎= to obtain the possible values of 𝑝 and 𝑞,

respectively. Be sure to consider both positive and negative values.

𝑝 = ±1,±2,±5,±10 𝑞 = ±1,±2

Step 4: List the possible values of U

V by forming all possible combinations of the

values of 𝑝 and 𝑞.

𝑝𝑞 = ±1,±2,±5,±10,±

12 ,±

52

Therefore, the possible rational zeros of 𝑓(𝑥) = 2𝑥) − 7𝑥- − 17𝑥 + 10 are

±1, ±2,±5,±10,± ?-, ± 3

-.

Let’s Practice!


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STUDY GUIDE

Try It Yourself! What are the possible rational zeros of 𝑓(𝑥) = 2𝑥- − 5𝑥 − 3? Example 2: What are the possible rational roots of −7𝑥- + 2𝑥4 + 9𝑥) = −5 + 9𝑥? Solution:

Step 1: Arrange the terms in decreasing order of degree. Make sure to equate the polynomial to zero.

−7𝑥- + 2𝑥4 + 9𝑥) = −5 + 9𝑥

−7𝑥- + 2𝑥4 + 9𝑥) − 9𝑥 + 5 = 02𝑥4 + 9𝑥) − 7𝑥- − 9𝑥 + 5 = 0

Step 2: Determine the values of 𝑎A and 𝑎=. In this case, 𝑎A = 5 and 𝑎= = 2. Step 3: List the factors of 𝑎A and 𝑎= to obtain the possible values of 𝑝 and 𝑞,


𝑝 = ±1,±5 𝑞 = ±1,±2



values of 𝑝 and 𝑞.

𝑝𝑞 = ±1,±5,±

12 ,±

52

Thus, the possible rational roots of −7𝑥- + 2𝑥4 + 9𝑥) = −5 + 9𝑥 are

±1,±5,± ?-, ± 3

-.


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STUDY GUIDE

Try It Yourself! What are the possible rational roots of 10𝑥- + 𝑥) + 23𝑥 = −14? Example 3: Find the polynomial 𝑃(𝑥) of degree 3 whose zeros are 2, −6, and −4

).

Solution: The Factor Theorem states that if 𝑘 is a zero of a given polynomial, then 𝑥– 𝑘

is a factor of that polynomial. Hence, the given zeros 2, −6, and −4)

correspond to the binomials 𝑥 − 2, 𝑥 + 6, and 3𝑥 + 4, respectively. Also, these expressions are factors of the required 𝑓(𝑥), which means that we can multiply these factors to obtain the polynomial.

𝑃(𝑥) = (𝑥 − 2)(𝑥 + 6)(3𝑥 + 4)

𝑃(𝑥) = (𝑥- + 4𝑥 − 12)(3𝑥 + 4) 𝑃(𝑥) = 3𝑥) + 16𝑥- − 20𝑥 − 48

Thus, the polynomial 𝑃(𝑥) whose zeros are 2, −6, and −4) is given by

3𝑥) + 16𝑥- − 20𝑥 − 48. What have you noticed about the number of zeros given and the degree of the polynomial we have obtained? There are three zeros, and the degree of the polynomial is 3. This is supported by the Fundamental Theorem of Algebra.

Try It Yourself!

Find the polynomial 𝑓(𝑥) of degree 3 whose zeros are 1, −2, and )4.


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STUDY GUIDE

Real-World Problems Example 4: The length of a storage compartment is

1 ft more than its height, and its width is 2 ft less than its height. If the volume of the compartment is 90 ft3, find the possible measurement of the height of the compartment.

Solution: The storage compartment is in the shape of a rectangular prism whose

volume 𝑉 is given by the formula


height. The expressions that represent the length, width, and height of the block are

𝑙 = 𝑥 + 1, 𝑤 = 𝑥 − 2, and ℎ = 𝑥, respectively. Substituting the given dimensions into the formula and multiplying the expressions results in the following solution:

𝑉 = 𝑙𝑤ℎ90 = (𝑥 + 1)(𝑥 − 2)(𝑥)90 = 𝑥) − 𝑥- − 2𝑥0 = 𝑥) − 𝑥- − 2𝑥 − 90

or 𝑥) − 𝑥- − 2𝑥 − 90 = 0

Since we are to find the possible measurement of the height of the

compartment, we are after the rational value/s of 𝑥 that will make the equation true. We use the Rational Root Theorem to find these possible rational values.


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STUDY GUIDE



𝑝 = ±1,±2,±3,±5,±6,±9,±10,±15,±18,±30,±45,±90 𝑞 = ±1



values of 𝑝 and 𝑞. Eliminate the negative values since we are only after positive ones which may represent the dimensions of the box.

𝑝𝑞 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Therefore, the possible measurement of the height of the compartment can

be any of the following values (in feet): 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. If we are after the exact measurement of the compartment’s height, we may test these values for a root of the given polynomial equation 𝑥) − 𝑥- − 2𝑥 − 90 = 0. This will be discussed in the next lesson.

Try It Yourself!

The length of a frame is one more than twice its width 𝑥. If the area of the frame is 36 square inches, what are the possible integral measurements the frame’s width?


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STUDY GUIDE

1. Find the possible rational roots of the following polynomial equations. a. 2𝑥) + 9𝑥- + 12𝑥 + 4 = 0 b. 𝑥) − 𝑥- + 7𝑥 = 7 c. 𝑥) + 𝑥- = 2 d. 𝑥4 − 2𝑥) + 2𝑥 − 4 = 0 e. 2𝑥3 − 3𝑥4 − 8𝑥) + 13𝑥- − 4 = 0

2. Find the polynomial equation with the following roots:

a. 1,−1, and 3 b. ?

-, −4 and 5

c. )4, − ?

- and −3

3. The volume of an ice cream cone is 147𝜋 cubic inches. If its radius is 2 units less

than the height 𝑥 of the cone, what are the possible measurements of the cone’s height?



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STUDY GUIDE

It’s a Lucky Find! Materials Needed: decoding activity sheet and pen Instructions: 1. Form 8 groups. Each group will be given the decoding activity sheet as shown

below. 2. Solve the given linear and quadratic equations. Each of these equations has its

corresponding letter. Match the letters to the solutions of the given equations. 3. The first group who successfully answered the question with correct solutions

wins the game.

What do you call the word that means a chance encounter with something wonderful? _______________

U: 5𝑥 − 6 = 14 R: 𝑥- − 2𝑥 − 15 = 0 E: 6𝑥 + 7 = 13 + 7𝑥 L: 4𝑥- − 100 = 0 T: 4𝑥 − 40 = 7(−2𝑥 + 2) O: (𝑥 − 4)- − 9 = 0 L: 7(5𝑥 − 4) − 1 = 14 − 8𝑥 V: 4𝑥- + 2𝑥 = 12 A: −8𝑥 + 4(1 + 5𝑥) = −6𝑥 − 14 I: 2𝑥- − 7𝑥 = 4

𝑥 = 3 𝑥 = 5,−3 𝑥 = 7, 1 𝑥 = 4 𝑥 =

32 ,−2 𝑥 = −1 𝑥 = −

12 , 4 𝑥 = 1 𝑥 = ±5 𝑥 = −6

Lesson 3: Solving Polynomial Equations

Warm Up!


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STUDY GUIDE

The equations you were able to solve in the Warm Up! activity are linear and quadratic which has a degree of 1 and a degree of 2, respectively. How do we solve polynomial equations of degree greater than 2? We have previously discussed how to translate real-life situations into polynomial equations. In Example 4 of Lesson 2, we constructed an expression that represents the volume of the storage compartment given ℎ = 𝑥, 𝑙 = 𝑥 + 1, and 𝑤 = 𝑥 − 2. If the volume of the compartment is 90 ft3, then its volume can be represented by 𝑥) − 𝑥- − 2𝑥 − 90 = 0. Moreover, we have identified that the possible measurement of the height of the compartment can be any of the following values (in feet): 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. If we are to determine the exact height, we need to learn the skill of solving polynomial equations. That is, we have to identify the value/s of 𝑥 that will make the equation true. The solution is stated as follows. Solution: The storage compartment is in the shape of a rectangular prism whose



height. The expressions that represent the length, width, and height of the block are

𝑙 = 𝑥 + 1, 𝑤 = 𝑥 − 2, and ℎ = 𝑥, respectively. Substituting the given dimensions into the formula and multiplying the expressions result in the following solution:

𝑉 = 𝑙𝑤ℎ

Learn about It!


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STUDY GUIDE

90 = (𝑥 + 1)(𝑥 − 2)(𝑥)90 = 𝑥) − 𝑥- − 2𝑥0 = 𝑥) − 𝑥- − 2𝑥 − 90

or 𝑥) − 𝑥- − 2𝑥 − 90 = 0

Since we are to find the possible measurement of the height of the

compartment, we are after the rational value/s of 𝑥 that will make the equation true. We use the Rational Root Theorem to find these possible rational values.



𝑝 = ±1,±2,±3,±5,±6,±9,±10,±15,±18,±30,±45,±90 𝑞 = ±1



values of 𝑝 and 𝑞. Eliminate the negative values since we are only after positive ones which may represent the dimensions of the box.

𝑝𝑞 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Step 4: Test these values for a zero of 𝑓(𝑥) = 𝑥) − 𝑥- − 2𝑥 − 90 by substituting these

to the value of 𝑥. Here are the solutions for some of the possible rational roots.

𝑓(𝑥) = 𝑥) − 𝑥- − 2𝑥 − 90 𝑓(1) = (1)) − (1)- − 2(1) − 90 𝑓(1) = 1 − 1 − 2 − 90


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STUDY GUIDE

𝑓(1) = −92

𝑓(𝑥) = 𝑥) − 𝑥- − 2𝑥 − 90 𝑓(2) = (2)) − (2)- − 2(2) − 90 𝑓(2) = 8 − 4 − 4 − 90 𝑓(2) = −90

𝑓(𝑥) = 𝑥) − 𝑥- − 2𝑥 − 90 𝑓(3) = (3)) − (3)- − 2(3) − 90 𝑓(3) = 27 − 9 − 6 − 90 𝑓(3) = −78

𝑓(𝑥) = 𝑥) − 𝑥- − 2𝑥 − 90 𝑓(5) = (5)) − (5)- − 2(5) − 90 𝑓(5) = 125 − 25 − 10 − 90 𝑓(5) = 0

Step 5: Since 𝑓(5) = 0, (𝑥 − 5) is a factor of 𝑥) − 𝑥- − 2𝑥 − 90. We divide this given

polynomial by 𝑥 − 5 to obtain the other factors. The result is 𝑥- + 4𝑥 + 18.

5 1 −1 −2 −90 5 20 90 1 4 18 0

We can describe the roots of 𝑥- + 4𝑥 + 18 by its discriminant. It is the number 𝐷 = 𝑏- − 4𝑎𝑐 determined from the coefficients of the equation 𝑎𝑥- + 𝑏𝑥 + 𝑐 = 0. If 𝐷 < 0, the roots are two complex conjugates. If 𝐷 = 0, there is only one root with a multiplicity of two. If 𝐷 > 0 that is a perfect square, then there are two distinct rational roots. Lastly, if 𝐷 > 0 that is not a perfect square, then there are two distinct irrational roots.

The discriminant of 𝑥- + 4𝑥 + 18 = 0 is −56. Thus, its roots are two complex

conjugates which cannot be the possible dimensions of the box.


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The only rational value of 𝑥 that made the equation true is 5. Therefore, the height 𝑥 of the storage compartment is 5 ft. Substitute 𝑥 = 5 to the expressions that represents the compartment’s width and length.

length = 𝑥 + 1 = 6 ft width = 𝑥 − 2 = 3 ft

The dimension of the compartment is 5 ft x 6 ft x 3 ft.

Example 1: Find the rational roots of 𝑓(𝑥) = 0 where 𝑓(𝑥) = 2𝑥) − 7𝑥- − 17𝑥 + 10. Solution:

Step 1: Write the equation in general form. The general form of a polynomial equation is given by

𝑎=𝑥= + 𝑎=>?𝑥=>? + ⋯+ 𝑎?𝑥 + 𝑎A = 0

where 𝑎A, 𝑎?,⋯ , 𝑎= are real numbers such that 𝑎= ≠ 0 and 𝑛 is a nonnegative

integer.

2𝑥) − 7𝑥- − 17𝑥 + 10 = 0

Step 2: Use the Rational Root Theorem. Determine the possible rational values of UV.

We have already accomplished this in Example 1 of Lesson 2. The possible

rational roots of 2𝑥) − 7𝑥- − 17𝑥 + 10 = 0 are ±1,±2,±5,±10,± ?-, ± 3

-.

Step 3: Test these values for a zero of 𝑓(𝑥) = 2𝑥) − 7𝑥- − 17𝑥 + 10 by substituting

these to the value of 𝑥. Here are the solutions for some of the possible rational roots.

Let’s Practice!


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STUDY GUIDE

𝑓(1) = 2(1)) − 7(1)- − 17(1) + 10 𝑓(1) = 2 − 7 − 17 + 10 𝑓(1) = −12

𝑓(−1) = 2(−1)) − 7(−1)- − 17(−1) + 10 𝑓(−1) = −2 − 7 + 17 + 10 𝑓(−1) = 18

𝑓(2) = 2(2)) − 7(2)- − 17(2) + 10 𝑓(2) = 16 − 28 − 34 + 10 𝑓(2) = −36

𝑓(−2) = 2(−2)) − 7(−2)- − 17(−2) + 10 𝑓(−2) = −16 − 28 + 34 + 10 𝑓(−2) = 0

Verify the rest of the possible roots using the same procedure, then compare

the function values you obtained with the ones in the following list.

𝑝𝑞 𝑓 `

𝑝𝑞a

𝑝𝑞 𝑓 `

𝑝𝑞a

5 0 12 0

−5 −330 −12

332

10 1140 52 −45

−10 −2520 −52 −45

The values −2, 5, and ?- made 𝑓(𝑥) = 0. Thus, the rational roots of 𝑓(𝑥) = 0

where 𝑓(𝑥) = 2𝑥) − 7𝑥- − 17𝑥 + 10 are −2, 5, and ?-.


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STUDY GUIDE

There is another way of getting the rational roots of the given equation without substituting all the possible values. Upon getting a value where 𝑓(𝑥) = 0, use the Factor Theorem to find the other factors of the given polynomial. We proceed with the first value we got which made 𝑓(𝑥) = 0. This value is −2.

Step 1: Since 𝑓(−2) = 0, (𝑥 + 2) is a factor of 2𝑥) − 7𝑥- − 17𝑥 + 10. We divide this

given polynomial by 𝑥 + 2 to obtain the other factors, which we can also call the depressed polynomials.

−2 2 −7 −17 10

−4 22 −10 2 −11 5 0

The depressed polynomial is 2𝑥- − 11𝑥 + 5. Step 2: The polynomial 2𝑥- − 11𝑥 + 5 can be further factored as (𝑥 − 5)(2𝑥 − 1). By

factoring 2𝑥- − 11𝑥 + 5, we obtain the complete factors of 𝑓(𝑥).

𝑓(𝑥) = 2𝑥) − 7𝑥- − 17𝑥 + 10𝑓(𝑥) = (𝑥 + 2)(𝑥 − 5)(2𝑥 − 1)

Step 3: Apply the zero-product property. The zero-product property states that if 𝑎𝑏 = 0, then either 𝑎 = 0 or 𝑏 = 0

(or both).

Whichever method is used, we were still able to obtain the same set of rational roots.


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STUDY GUIDE

Try It Yourself! What are the rational roots of 𝑓(𝑥) = 0 where 𝑓(𝑥) = 𝑥) + 𝑥- − 16𝑥 − 16? Example 2: Solve the polynomial equation 𝑥4 − 4𝑥) + 3𝑥- + 4𝑥 − 4 = 0. Solution:

Step 1: Write the equation in general form. The given equation is already in general form, so we proceed to the next step right away.

Step 2: Factor the polynomial. Use the Rational Root Theorem to find the possible

roots of the given equation. To determine U

V, list the factors of 𝑎A = −4 and 𝑎= = 1.

𝑝 = ±1,±2,±4

𝑞 = ±1 𝑝𝑞 = ±1,±2,±4

Check if 1 is a root of the equation. 𝑃(𝑥) = 𝑥4 − 4𝑥) + 3𝑥- + 4𝑥 − 4

𝑃(1) = (1)4 − 4(1)) + 3(1)- + 4(1) − 4 𝑃(1) = 1 − 4 + 3 + 4 − 4 𝑃(1) = 0

Since 1 is a root, 𝑥 − 1 is a factor of the polynomial 𝑥4 − 4𝑥) + 3𝑥- + 4𝑥 − 4. Divide 𝑥4 − 4𝑥) + 3𝑥- + 4𝑥 − 4 by 𝑥 − 1 to find the depressed polynomial.


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STUDY GUIDE

1 1 −4 3 4 −4 1 −3 0 4 1 −3 0 4 0

The depressed polynomial is 𝑥) − 3𝑥- + 4. Check another root of the equation to get a factor of 𝑥4 − 4𝑥) + 3𝑥- + 4𝑥 − 4. Check if −1 is a root of the equation.

𝑃(𝑥) = 𝑥4 − 4𝑥) + 3𝑥- + 4𝑥 − 4𝑃(−1) = (−1)4 − 4(−1)) + 3(−1)- + 4(−1) − 4𝑃(−1) = 1 + 4 + 3 − 4 − 4𝑃(−1) = 0

Since −1 is a root, 𝑥 + 1 is a factor of the polynomial 𝑥4 − 4𝑥) + 3𝑥- + 4𝑥 − 4. Divide the depressed polynomial 𝑥) − 3𝑥- + 4 by 𝑥 + 1, then factor the

quotient.

−1 1 −3 0 4 −1 4 −4 1 −4 4 0

The quotient is 𝑥- − 4𝑥 + 4. The factors of this polynomial are (𝑥 − 2)(𝑥 − 2). Thus, 𝑥4 − 4𝑥) + 3𝑥- + 4𝑥 − 4 = (𝑥 − 1)(𝑥 + 1)(𝑥 − 2)(𝑥 − 2).

Step 3: Apply the zero-product property.

𝑥4 − 4𝑥) + 3𝑥- + 4𝑥 − 4 = 0(𝑥 − 1)(𝑥 + 1)(𝑥 − 2)(𝑥 − 2) = 0


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STUDY GUIDE

Therefore, the solutions of the equation 𝑥4 − 4𝑥) + 3𝑥- + 4𝑥 − 4 = 0 are

1,−1, 2, and another 2. Since there are two 𝑥 − 2 factors, 2 is called a multiple root of the polynomial equation.

Try It Yourself! What are the solutions of the equation 𝑥4 + 2𝑥) − 8𝑥- − 18𝑥 − 9 = 0? Example 3: What is the multiple root of 2𝑥) + 7𝑥- + 4𝑥 − 4 = 0?

Solution: The multiple root is the root of the polynomial equation that appears more

than once.

Step 1: Write the equation in general form. The given equation is already in general form, so we proceed to the next step right away.

Step 2: Factor the polynomial. Use the Rational Root Theorem to find the possible

roots of the given equation. To determine U


𝑝 = ±1,±2,±4

𝑞 = ±1,±2 𝑝𝑞 = ±1,±2,±4,±

12

Check if −2 is a root of the equation.


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STUDY GUIDE

𝑃(𝑥) = 2𝑥) + 7𝑥- + 4𝑥 − 4𝑃(−2) = 2(−2)) + 7(−2)- + 4(−2) − 4𝑃(−2) = −16 + 28 − 8 − 4𝑃(−2) = 0

Since −2 is a root, 𝑥 + 2 is a factor of the polynomial 2𝑥) + 7𝑥- + 4𝑥 − 4. Divide 2𝑥) + 7𝑥- + 4𝑥 − 4 by 𝑥 + 2 to find the depressed polynomial.

−2 2 7 4 −4 −4 −6 4 2 3 −2 0

The quotient is 2𝑥- + 3𝑥 − 2. The factors of this polynomial are

(𝑥 + 2)(2𝑥 − 1). Thus, 2𝑥) + 7𝑥- + 4𝑥 − 4 = (𝑥 + 2)(𝑥 + 2)(2𝑥 − 1). Step 3: Apply the zero-product property.

2𝑥) + 7𝑥- + 4𝑥 − 4 = 0(𝑥 + 2)(𝑥 + 2)(2𝑥 − 1) = 0

Since there are two 𝑥 + 2 factors, −2 is the multiple root of the polynomial equation.

Try It Yourself!

What is the multiple root of the equation 𝑥) + 5𝑥- − 25𝑥 − 125?


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STUDY GUIDE

Real-World Problems Example 4: The length of a frame is one more than twice its width 𝑥. If the

area of the frame is 36 square inches, what are the dimensions of the frame?

Solution: The frame is in the shape of a rectangle whose area is given by

the formula

𝐴 = 𝑙𝑤 where 𝑙 is the length of the rectangle, and 𝑤 is its width. The expression that

represents the length and width of the frame are 𝑙 = 2𝑥 + 1 and 𝑤 = 𝑥, respectively. Substituting the given dimensions into the formula and multiplying the expressions results in the following solution:

𝐴 = 𝑙𝑤36 = (2𝑥 + 1)(𝑥)36 = 2𝑥- + 𝑥0 = 2𝑥- + 𝑥 − 36

or 2𝑥- + 𝑥 − 36 = 0

Since we are to find the dimensions of the frame, we are after the rational

value/s of 𝑥 that will make the equation true. We use the Rational Root Theorem to find these possible rational values.

Step 1: Write the equation in general form. The given equation is already in general

form, so we proceed to the next step right away.


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STUDY GUIDE

Step 2: Factor the polynomial. Use the Rational Root Theorem to find the possible roots of the given equation. Eliminate the negative values since we are only after positive ones which may represent the dimensions of the frame.

To determine U


𝑝 = 1, 2, 3, 4, 6, 9, 12, 18, 36 𝑞 = 1, 2

𝑝𝑞 = 1, 2, 3, 4, 6, 9, 12, 18, 36,

32 ,92

Check if 4 is a root of the equation.

𝑃(𝑥) = 2𝑥- + 𝑥 − 36𝑃(4) = 2(4)- + 4 − 36𝑃(4) = 32 + 4 − 36𝑃(4) = 0

Since 4 is a root, 𝑥 − 4 is a factor of the polynomial 2𝑥- + 𝑥 − 36. The other

factor of this polynomial is (2𝑥 + 9). Thus, 2𝑥- + 𝑥 − 36 = (𝑥 − 4)(2𝑥 + 9).


2𝑥- + 𝑥 − 36 = 0(𝑥 − 4)(2𝑥 + 9) = 0


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We can only use 4 inches as the frame’s width since a negative value of measurement cannot be possible. To solve for the length of the frame, we substitute 𝑥 = 4 to the expression that represents it.

𝑙 = 2𝑥 + 1 𝑙 = 2(4) + 1

𝑙 = 9 inches The length and width of the frame is 9 inches and 4 inches, respectively.

Try It Yourself!

The width of a book’s cover is 5 less than its length. If the area of the book cover is 104 in2, what are its dimensions?

1. Solve the following polynomial equations. a. 2𝑥) + 9𝑥- + 12𝑥 + 4 = 0 b. 𝑥) − 𝑥- + 7𝑥 = 7 c. 𝑥) + 𝑥- = 2 d. 𝑥4 − 2𝑥) + 2𝑥 − 4 = 0 e. 2𝑥3 − 3𝑥4 − 8𝑥) + 13𝑥- − 4 = 0

2. Find the multiple root of the following polynomial equations.

a. 𝑥) + 5𝑥- − 25𝑥 − 125 = 0 b. 4𝑥- + 4𝑥 + 1 = 0 c. 𝑥) + 𝑥- − 𝑥 − 1 = 0



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STUDY GUIDE

Place your Bet! Materials Needed: whiteboard, whiteboard marker, papers for solutions Instructions: 1. Form 8 groups. 2. This game assesses what you have learned from the previous lessons. Your

teacher will prepare the following questions: 2 Easy Questions: Translating Real-life Situations into Polynomial Equations 2 Average Questions: Rational Root Theorem 2 Difficult Questions: Solving Polynomial Equations

3. Before the question is given, you are to place your bets (100 points as the highest bet and 0 point for the lowest bet).

4. If you answered the question correctly, you get the point that you decided to bet. Otherwise, that same point will be deducted from your score.

5. All groups start from 0 point. If you bet 100 points to the first question and you were able to answer it correctly, then you get 100 points. Otherwise, you get a negative 100.

6. The game ends after the 6 questions. The group who was able to gather the highest point wins the game.

Lesson 4: Solving Problems Involving Polynomial Equations

Warm Up!


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STUDY GUIDE

Using the concepts discussed in the previous lessons, let us try to solve some word problems involving polynomial equations.

Example 1: A wooden rectangular box has a volume of 96in3. Its length is 8 inches more

than its height and its width is 2 inches less that its height. Express the volume of the box in terms of its height 𝑥.

Solution:

Step 1: Identify the working formula. The wooden rectangular box is in the shape of a rectangular prism whose



height.

Step 2: Express all dimensions in terms of the height of the box.

Height: ℎ Width: ℎ + 8 Length: ℎ − 2

Let’s Practice!

Learn about It!


40

STUDY GUIDE

Step 3: Set up the algebraic model and write the polynomial equation in standard form.

Substituting the given dimensions into the formula to get the volume of a

rectangular prism, and multiplying the expressions, we get the following solution:

𝑉 = 𝑙𝑤ℎ96 = (ℎ − 2)(ℎ + 8)(ℎ)96 = ℎ) + 6ℎ- − 16ℎ

Thus, the volume of the wooden rectangular box is represented by the

equation 96 = ℎ) + 6ℎ- − 16ℎ.

Try It Yourself!

A cylindrical tank has a height of one more than thrice its radius. What algebraic equation will represent the volume of the tank if its capacity is 90𝜋 ft3?

Example 2: A museum intends to put one of its artifacts on display in a pyramid-shaped

glass case. The height of the case must be thrice as tall as the artifact and the edge of its square base is 5 ft shorter than the artifact’s height. If the volume of the pyramid is 250 ft3, find its dimensions.

Solution:

Step 1: Identify the working formula. The glass case is in the shape of a square pyramid whose volume 𝑉 is given

by the formula


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STUDY GUIDE

𝑉 =13𝑏

-ℎ

where 𝑏 is the base of the pyramid and ℎ is its height.

Step 2: Express all dimensions of the pyramid in terms of the height 𝑥 of the artifact.

Height: 3𝑥 Base: 𝑥 − 5


Substituting the given dimensions into the formula to get the volume of a

pyramid, and multiplying the expressions, we get the following solution:

𝑉 =13𝑏

-ℎ

250 =13(𝑥 − 5)-(3𝑥)

250 =13(𝑥- − 10𝑥 + 25)(3𝑥)

250 =13(3𝑥) − 30𝑥- + 75𝑥)

250 = 𝑥) − 10𝑥- + 25𝑥𝑥) − 10𝑥- + 25𝑥 − 250 = 0

Step 4: Factor by grouping.

𝑥) − 10𝑥- + 25𝑥 − 250 = 0𝑥-(𝑥 − 10) + 25(𝑥 − 10) = 0

(𝑥- + 25)(𝑥 − 10) = 0



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STUDY GUIDE

Note that dimensions cannot take complex values. So, the only real solution for this problem is 10.

Step 6: Substitute the solution to the dimensions expressed in algebraic form.

Height: 3𝑥 = 3(10) = 30 Base: 𝑥 − 5 = 10 − 5 = 5

Therefore, the pyramid should have a square base measuring 5ft x 5ft and a

height of 30ft.

Try It Yourself! A cylindrical tank has a height of one more than thrice its radius. If the capacity of

the tank is 90𝜋 ft3, then what is the measurement of its radius and height? Example 3: A rectangle of width 5𝑥 has an area given by 15𝑥- + 40𝑥. If the length of the

rectangle is 14 meters, find the measure of its width.

Solution:

Step 1: Identify the working formula. The area of a rectangle is


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STUDY GUIDE

𝐴 = 𝑙𝑤

where 𝑙 is the length and 𝑤 is the width of the rectangle.


Substituting the given dimensions into the formula to get the area of a

rectangle, and multiplying the expressions, we get the following solution:

𝐴 = 𝑙𝑤15𝑥- + 40𝑥 = (14)(5𝑥)15𝑥- + 40𝑥 = 70𝑥

15𝑥- + 40𝑥 − 70𝑥 = 015𝑥- − 30𝑥 = 0

Step 3: Factor by grouping.

15𝑥- − 30𝑥 = 015𝑥(𝑥 − 2) = 0


Note that dimensions cannot take zero values. So, the only real solution for this problem is 2.

To find the width, substitute 𝑥 = 2 to the expression 5𝑥. Therefore, the width

of the rectangle is 10 meters.


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STUDY GUIDE

Try It Yourself!

The area of a rectangle is given by the expression (31𝑥 + 6) square meters. If the width of the rectangle is 3𝑥 and its length is 4𝑥 − 1, then what are the dimensions of the rectangle?

More Real-World Problems Example 4: The equation for free fall on a moon of a distant

planet, in feet, is given by the equation ℎ(𝑡) = −𝑡- + 𝑉A𝑡 + ℎA, where ℎ(𝑡) is the height, 𝑉A is the initial velocity, ℎA is the initial height, and 𝑡 is the time. Suppose an astronaut dropped a ball off the rim of a crater 200 ft above its floor. If the initial velocity of the ball is 10 feet per second, when will the ball hit the crater’s bottom?

Solution: When the ball hits the crater’s bottom, its height ℎ(𝑡) = 0.

Step 1: Substitute the values in the given formula.

ℎ(𝑡) = −𝑡- + 𝑉A𝑡 + ℎA0 = −𝑡- + 10𝑡 + 2000 = 𝑡- − 10𝑡 − 200

Step 2: Factor the polynomial.

𝑡- − 10𝑡 − 200 = 0(𝑡 − 20)(𝑡 + 10) = 0


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STUDY GUIDE


Note that dimensions cannot take negative values. So, the only real solution

for this problem is 20. The ball will hit the bottom of the crater 20 seconds after it is dropped.

Try It Yourself!

A specific drug for curing a virus is injected into the bloodstream of a person. The rate at which the virus decrease in number after 𝑡 hours is described by the function 𝑓(𝑡) = 𝑡4 − 𝑡) − 16𝑡- − 4𝑡 + 80. After how many hours will the virus be eradicated?

1. The sum of two numbers is 8. The sum of their squares is 40. Find the two numbers.

2. The length of a rectangle is 2 more than its width. If the area of the rectangle is 168 square centimeters, find its perimeter.

3. The sum of the squares of two consecutive odd numbers is 130. What are the two

odd numbers?



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STUDY GUIDE

4. The volume of a box is 192 square meters. What are the dimensions of the box if these are three consecutive even integers?

5. The width of a storage compartment is 8ft more than its height, and its length is 5ft

less than its height. If the volume of the compartment is120ft3, then find the dimensions of the storage.

1. What is the relationship between the factors of a polynomial and the roots of polynomial equations?

2. Find the value of 𝑘 in 𝑥) + 𝑘𝑥- − 16𝑥 − 112 so that (𝑥 + 4) is a factor of the given polynomial.

3. The polynomial equations 𝑥) + 𝑎𝑥- − 13𝑥 − 12 = 0 and 𝑥) + 𝑏𝑥- − 4𝑥 + 16 = 0 have a common solution of 4. Find the values of 𝑎 and 𝑏.

4. The polynomial 2𝑥) − 𝑎𝑥- − 𝑏𝑥 − 12 is exactly divisible by 𝑥- − 𝑥 − 12. Find the value of 𝑎 + 𝑏.

5. Is −2 a solution to the polynomial equation 𝑥?AA − 2?AA?

Challenge Yourself!


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STUDY GUIDE

You are a pastry designer trying out new pastries to sell in a bakery. You want your pastries to come in different sizes: rectangular prism, pyramidal, and cylindrical. Design the pastry and determine its dimensions. For each shape, assign a dimension and label it as 𝑥. Write expressions in terms of 𝑥 to represent the other dimensions. Finally, write the polynomial equation that represents the volume of each pastry you designed. Make a portfolio to present your output.

The owner of the bakery will use your design and information for further study and recommendation so make sure that it shows correct computation, creative design of the pastries, and is neat and organized. Performance Task Rubric

Criteria Below

Expectation (0–49%)

Needs Improvement

(50–74%)

Successful Performance

(75–99%)

Exemplary Performance

(99+%)

Accuracy of the Content

There are a significant number of errors in computations.

There are a few errors in the computation.

All computations are correct.

All computations are correct and with complete solution.

Creativity

The pastries are not aesthetically designed.

The designs of the pastries are clean and good, but may be improved further.

The designs of the pastries are creative and clean.

The pastries are aesthetically designed, neat, and organized.

Overall Presentation

The portfolio is not organized

The portfolio is organized but with some

The portfolio is neat. All information

The portfolio is organized properly. All

Performance Task


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STUDY GUIDE

properly. missing information.

needed is present.

information needed is present.

TOTAL

Wrap-up


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STUDY GUIDE

Key Concepts & Descriptions

Concepts Descriptions

Polynomial

A polynomial is an expression that contains variables and constants which can be combined using addition, subtraction, and multiplication. A polynomial in 𝑥is an expression written in the form

𝑎=𝑥= + 𝑎=>?𝑥=>? + ⋯+ 𝑎?𝑥 + 𝑎A

where 𝑎A, 𝑎?,⋯ , 𝑎= are real numbers such that 𝑎= ≠ 0 and 𝑛 is a nonnegative integer.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial function 𝑓(𝑥) of degree 𝑛 has exactly 𝑛 complex zeros. This theorem will help you check if you have the correct number of zeros.

Factor Theorem

If a polynomial function 𝑓(𝑥) is divided by a divisor 𝑥 − 𝑘 such that 𝑓(𝑘) = 0, then 𝑥 − 𝑘 is a factor of 𝑓(𝑥). However, if a polynomial is equated to zero, how do we find the possible values of 𝑥(called roots) that make the equation true? In general form, we write this as

𝑎=𝑥= + 𝑎=>?𝑥=>? + ⋯+ 𝑎?𝑥 + 𝑎A = 0 where 𝑎A, 𝑎?,⋯ , 𝑎= are real numbers such that 𝑎= ≠ 0 and 𝑛 is a nonnegative integer.

Rational Root Theorem The Rational Root Theorem states that the possible rational root of a polynomial


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STUDY GUIDE

equation 𝑓(𝑥) = 0 is of the form UV, where

the integer 𝑝 is a factor of the constant term𝑎A and the integer 𝑞 is a factor of the leading coefficient 𝑎=.

Lesson 1

1. 𝐴 = (𝑥- − 2𝑥) square units 2. 𝐴 = 2𝑥- square units

3. 𝑉 = cd√)4

cubic units

4. 𝑉 = (4𝑥) − 32𝑥- + 64𝑥) cubic inches Lesson 2

1. ±1,±3,± ?-, ± )

-

2. ±1,±2,±7,±14 3. 𝑓(𝑥) = 4𝑥) + 𝑥- − 11𝑥 + 6 4. 1, 2, 3, 4, 6, 9, 12, 18, 36

Lesson 3

1. 𝑥 = −4,−1, 4 2. 𝑥 = −1 (multiplicity of 3), 𝑥 = −3, 𝑥 = 3 3. 𝑥 = −5 (multiplicity of 2) 4. length = 13 inches, width = 8 inches

Lesson 4

1. 𝑉 = (3𝑥) + 𝑥- − 90) cubic feet 2. height = 10 inches, radius = 3 inches 3. width = 9 meters, length = 11 meters 4. t = 4 hours

Key to Let’s Practice!


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STUDY GUIDE

Brownmath. “Solving Polynomial Equations”. Accessed April 10, 2018.

https://brownmath.com/alge/ polysol.htm Mathwords. “Rational Root Theorem”. Accessed April 10, 2018.

http://www.mathwords.com/r/rational_ root_theorem.htm Mathspace. “Applications of Polynomials”. Accessed April 10, 2018.

https://mathspace.co/learn/world-of-maths/functions-graphs-and-behaviour/applications-of-polynomials-33940/applications-of-polynomials-1489/

References

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