Pre-Algebra 2

Unit 9 Polynomials

Name________________________

Period______

9.1A Add, Subtract, and Multiplying Polynomials (non-complex)

Explain

Add the following polynomials: 1) ( ) ( )

2) ( ) ( )

Subtract the following polynomials: 1) ( ) ( )

2) ( ) ( )

FOIL Method for Multiplying Binomials FOIL is an acronym for First, Outer, Inner, Last. Multiply these terms together and then

find their sum. You will notice that the box method and FOIL represent the same data.

Example:

532 xx

Total Area: Total Area:

1) Use FOIL to multiply the following polynomials.

5232 2 xxx Total Area:

2) Now, use the box method to multiply the polynomials

5232 2 xxx

Total Area:

2x + 3

x

-5 F L

O

I

Place the letters F, O, I, and L in the boxes to match

the same multiplication as in the FOIL example.

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3) What does (2x)2 look like when it’s written in expanded form?

4) What does (2x + 3)2 written in expanded form?

5) Draw your own box to find the simplified form of (2x + 3)2.

Multiply the following polynomials:

1) ( )( )( )

2) ( )( )( )

3) ( )( )( )

Extra:

One factor of ( ) is ( ). What are the other two factors?

Area Problems

1. Find the area. 2. Find the area.

2. Find the width. 6. Find the length.

yx36

478 yx 354 yx

yx319

4336 yxA

79xy

1154 yx yxA 352?

?

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9.1A Add, Subtract, and Multiply Polynomials WS

Simplify the following polynomials 1) ( ) ( )

2) ( ) ( )

3) ( ) ( )

Use the FOIL method to find the area of a rectangle with the following dimensions:

4) (2x – 4)(3x – 7) 5) (x – 4y)(-2x -5y) 6) (3x + 5)(x + 4)

Use the box method to simplify the following:

7) (2x)(-4x + 8) 8) (3x – 4)2 9) (3x + 4)(3x – 4)

Use the box to find the missing amounts:

10) ( )( ) )

11) ( )( ) )

Use a box to determine the dimensions that would create the following area:

12) X2 + 5x 6

13) 25x2 + 40x + 16

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1. Please find the area. 2. Please find the area.

3. Please find the length. 4. Please find the width.

zyx 837

46215 zyx

922x

419 zy

537 yzx 362105 zyxA 3919 zy

9257 zyA ?

?

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9.1B Add, Subtract, and Multiplying Complex Numbers

Explain

Add the following polynomials: 1) ( ) ( ) ( )

2) ( ) ( ) ( )

Subtract the following polynomials:

1) ( ) ( )

2) ( ) ( ) ( )

Multiply the following polynomials:

1) ( )( )

2) ( )( )

3) ( )( )

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9.1B Add, Subtract, and Multiplying Complex Numbers WS

Add the following polynomials:

1) ( ) ( ) ( )

2) ( ) ( )

Subtract the following polynomials:

1) ( ) ( )

2) ( ) ( ) ( )

Multiply the following polynomials: 1) ( )( )

2) ( )( )

3) ( )( )

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9.2 Notes: Long Division of Polynomials Name _____________________ Algebra II Date ____________ Per ______ Long Division Steps:

1. Divide the first term of the dividend by the first term of the divisor. 2. Write the result from step 1 in the quotient and use it to multiply the divisor. 3. Subtract the product from the dividend. 4. Repeat steps 1-3 using the difference from step 3 as the new dividend.

Example: Find the quotient. 58964 ÷ 5 Example: Find the quotient. 58964 ÷ 25

Example: Find the quotient. 3 2 23 3 2 1x x x x x

Dividend = 3 23 3 2x x x

Divisor = 2 1x x

2 3 2

2

1 3 3 2

x

x x x x x

3 2x x x

22 2 2x x

22 2 2x x

0 remainder

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Use long division.

1. 2( 5 14) ( 2)x x x 2. 2( 2 48) ( 6)x x x

3. 3 2( 3 16 12) ( 1)x x x x 4. 3 2( 3 8 5) ( 1)x x x x

5. 4 2( 7 9 10) ( 2)x x x x 6. 3 2 2( 21 45) ( 2 15)x x x x x

7. 3 2 2(8 5 12 10) ( 3)x x x x 8. 4 3 2(4 2 9 12) ( 2 )x x x x x

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9.2 WS: Long Division of Polynomials Name _____________________ Algebra II Date ____________ Per ______ Use long division.

1. 2( 6 8) ( 4)x x x 2. 2(2 7 10) ( 2)x x x

3. 3 2( 10 19 30) ( 6)x x x x 4. 3( 4 6) ( 3)x x x

5. 4 2(4 15 4) ( 2)x x x 6. 3 2 2(3 11 4 1) ( )x x x x x

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7. 4 3 2( 5 8 13 12) ( 6)x x x x x 8. 3 2(3 34 72 64) (3 2)x x x x

9. 3 2 2(7 11 7 5) ( 1)x x x x 10. 4 3 2 2(2 3 2 4) ( 1)x x x x x x

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9.3 Notes: Synthetic Division of Polynomials Name _____________________ Algebra II Date ____________ Per ______ Synthetic Division Steps:

1. Write the coefficients of the polynomial and then write the value of r on the left. Write the first coefficient below the line.

2. Multiply the r-value by the number below the line, and write the product below the next coefficient.

3. Write the sum (not the difference) below the line. Multiply r by the number below the line and write the product below the next coefficient.

4. Write the sum (not the difference) below the line. Repeat steps 1-3 as needed.

Note: Synthetic Division can only be used on linear divisors (i.e. in the form x - r). If the divisor is in any

other form, Long Division must be used.

Use synthetic division.

1. 2( 5 6) ( 1)x x x 2. 3( 30) ( 3)x x x

3. 4 3 2(2 11 15 6 18) ( 3)x x x x x 4. 2( 2 48) ( 5)x x x

5. 4 2( 7 9 10) ( 2)x x x x 6. 3 2(3 16 103 36) ( 4)x x x x

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Synthetic division can be used to factor. In example 2, x3 + x + 30 was divided by x + 3 and the

answer was _____________________.

Thus, x3 + x + 30 = ( )( )

7. Factor f(x) = x3 – 18x2 + 95x – 126 given that x – 9 is a factor.

8. Factor f(x) = 2x3 + 3x2 – 39x – 20 given that -5 is a zero of f(x).

Remainder & Factor Theorems

Remainder Theorem

If a polynomial f(x) is divided by x – a, the remainder is the constant f(a), and:

f(x) = q(x) • (x – a) + f(a)

where q(x) is a polynomial with degree one less that the degree of f(x).

9. Let f(x) = 2x4 – x3 + 4. Show that f(-1) is the remainder when f(x) is divided by x + 1.

10. Let f(x) = x3 + 5x2 – 7x + 2. Find f(2).

Factor Theorem

Let a polynomial f(x) be divided by x – a. If the remainder is 0, then x – a is a factor of f(x).

11. Let f(x) = 3x3 – 4x2 – 28x – 16. Is x + 2 a factor?

( )

( )

( )

q x

x a f x

WORK

f a

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9.3 WS: Synthetic Division of Polynomials Name _____________________ Algebra II Date ____________ Per ______ Use synthetic division.

1. 2( 7 12) ( 3)x x x 2. 2(4 13 5) ( 2)x x x

3. 3( 4 6) ( 3)x x x 4. 3 2( 5 2) ( 4)x x x

5. 3 2( 6 5 12) ( 4)x x x x 6. 3 2( 18 95 150) ( 10)x x x x

7. 4 3 2( 5 8 13 12) ( 6)x x x x x 8. 3 2(4 27 3 64) ( 7)x x x x

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9. 4 3 2( 4 13 4 12) ( 1)x x x x x 10. 4 3( 6 40 33) ( 7)x x x x

Use the Remainder Theorem to determine the following. 11. Let f(x) = 3x4 – 2x3 + x – 3. Find f(-1). 12. Let f(x) = x3 + 2x2 – 5x – 2. Find f(3). Use the Factor Theorem to determine the following. 13. Let f(x) = x4 – 4x3 – 20x2 + 48x. Is (x + 4) a factor? 14. Let f(x) = x3 – x2 – 5x + 3. Is (x – 3) a factor?

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9.4 Solving by Factoring (grouping)

Elaborate Activity 1. Write the standard form for a quadratic trinomial: ________________________

2. Given the trinomial, 2x 2 5x 12, we will express this as the product of two binomials. Identify A________ Identify B________ Identify C________ Our job will be easier if we can rewrite this trinomial as a polynomial with 4 terms. Then we can group the terms in pairs and use GCF factoring!!!!! 3. When we multiply the coefficients of the first and last terms, AC, we have our target product.

2x2 5x 12 2 times -12 equals -24 AC, our target product = _______. Our target sum, the coefficient of the middle term, B, = _______. We need to find factors of the target product, -24, that add to equal the target sum, -5, in order to begin factoring our trinomial.

4. Using, 2x2 5x 12, find the pair of factors for AC, our target product that we need for our target sum, -5. Since AC = -24, let’s check some factors and their sums in the following chart:

Factors of -24 Sum

-1 24 23

-2 12

-3 8

-4 6 2

-6 4

-8 3

-12 2 -10

-24 1

Highlight the one which has the target product of -24 and target sum of -5.

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5. Let’s rewrite 2x 2 5x 12 as a polynomial with 4 terms, using our target factors of –8 and 3.

2x2 5x 12 Highlight what changed.

2x2 8x + 3x 12 We have only changed the form of the original trinomial but the polynomial is the same.

6. 2x2 5x 12

2x2 8x + 3x 12 By grouping (circle the first two and second two together) these 4 terms in pairs, we can factor out the GCF from each pair.

2x2 8x + 3x 12

____ (x 4) + ____(x 4)

Write these two terms as one binomial factor, (2x + 3) and the binomial (x 4) is our other factor.

So, 2x 2 5x 12 in factored form, is (2x + 3)(x 4).

Check by multiplying (2x + 3)(x 4) to see if the result is 2x 2 5x 12.

7. Let’s factor another trinomial. Factor 4x2 9x + 5, using the AC method we just learned. AC, our target product = _______. Our target sum = _______. We need to find factors of the target product, _____ that add to equal the target sum, _____.

Factors of 20 Addition of Factors Sum of Factors Result?

Rewrite 4x2 9x + 5 as a polynomial with 4 terms, using our target factors of _____ and _____.

So, 4x2 9x + 5 in factored form, is ( )( ).

8. Factor 3x2 + 22x + 7. target product = _______ target sum = _______

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Evaluate (Assignment) Factor completely.

1. 2y 2+ 7y + 5 2. r 2– 20r + 36

3. 4x 2+ 7x + 3 4. 12a 2+ 10a – 8

5. d 2+ 4d – 21 6. p 2+ 14p + 49

7. 4x 2+ 11x + 6 8. 3q 2– q – 2

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9. 5x 2– 3x – 17 10. w 3+ 2w 2– 35w

11. 4h 2+ 8h – 42 12. k 2+ 12k + 36

13. 2x 2 + 7x + 3 14. 3x 2+ 13x + 4

15. 2y 3– 8y 2– 42y

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9.5 Notes: Polynomial Behavior PAP Algebra II Name ______________________ Date __________ Per ____

Function

Factored Form Graph of f(x)

Root (x = )

Crosses, Tangent, Wiggles

&

Power of Factor

End Behavior

Term of

Highest

Degree &

Odd or Even

x = -2 Left Right nax

1. ( ) ( 2)( 1)( 3)f x x x x

2. 2( ) ( 2)( 2) ( 6)f x x x x

3. 2 2( ) 2( 2)( 5) ( 3)f x x x x

4. 3( ) 2( 2)( 1) ( 1)f x x x x

5. 2 3( ) ( 2) ( 1) ( 4)f x x x x

1. Graph each function on your graphing calculator.

2. Sketch the graph on the chart. (Do not worry about scale. We are just interested in x-intercepts and end behavior.)

3. Fill in the remaining columns of the chart based on your graph.

4. What does the power of a factor tell you about how f(x) crosses the x-axis at the corresponding root?

5. If f(x) has a highest powered term of nax , how does its end behavior look if:

a > 0, n even: __________ a > 0, n odd: __________ a < 0, n even: __________ a < 0, n odd: __________ 19

9.5 WS: Polynomial Behavior PAP Algebra II Name _______________________ Date __________ Per ____

Sketch the graph on the chart. (Do not worry about scale. We are just interested in x-intercepts and end behavior.)

Fill in the remaining columns of the chart based on your graph.

Function Graph of f(x)

Crosses, Tangent, Wiggles

&

Power of Factor

End Behavior Term of

Highest

Degree

x = -2 x = 1 x = 3 Left Right nax

1. 3 2( ) ( 2) ( 1)( 3)f x x x x

Wiggles

3

Crosses

1

Tangent

2 Up Up x 61

2. 2 2( ) ( 2) ( 1)( 3)f x x x x

3. 3 2( ) ( 2)( 1) ( 3)f x x x x

4. 4 2( ) ( 2)( 1) ( 3)f x x x x

5. 2( ) ( 2) ( 1)( 3)f x x x x

6. 5( ) ( 2) ( 1)( 3)f x x x x

7. 2( ) 2( 2)( 1) ( 3)f x x x x

8. 3( ) 2( 1)f x x

1 3 -2

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