topic 3: polynomial functions
TRANSCRIPT
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Topic 3 Note Packet
Algebra 2 Name:________________________________________
Period:______
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Topic 3: Polynomial Functions
Date Section Topic HW Due Date
3.1 Graphing Polynomial Functions
3.2 Adding, Subtracting, and Multiplying Polynomials
3.3 Polynomial Identities
3.4 Dividing Polynomials
3.5 Zeros of Polynomials Functions
3.6 Theorems About Roots of Polynomial Equations
3.7 Transformations of Polynomial Functions
Quiz
Quiz
Quiz
Review
Test
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3.1 Graphing Polynomial Functions
Ex 1 How can you write a polynomial in standard form and use it to identify the leading coefficient, the degree, and
the number of terms?
Standard Form Leading Coefficient Degree
Try It! What is each polynomial in standard form and what are the leading coefficient, the degree, and the number
of terms of each?
a. b.
Ex 2 How do the sign of the leading coefficient and degree of a polynomial affect the end behavior of the graph of a
polynomial function?
Try It! Use the leading coefficient and degree of the polynomial function to determine the end behavior of each
graph.
a. b.
Ex 3 Consider the polynomial function
A. How can you use a table of values to identify key B. How can you use the graph to estimate the average
rate of features and sketch a graph of the function? change over the interval [-2, 0]?
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Try It! Consider the polynomial function
a. Describe the key features. Fill in the table b. Find the average rate of change of change over the
of values, and graph. interval [0, 2].
Ex 4 How can you sketch a graph of the polynomial function f from a verbal description?
Try It! Use the information below to sketch the graph of π¦ = π(π₯)
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Ex 5 In science class, Abby mixes a fixed amount of baking soda with different amounts of vinegar in a bottle
capped by a balloon. She records the amount of time it takes the gases produced by the reaction to inflate the balloon.
From her data, Abby created a function to model the situation. For x quarter-cups of vinegar, it takes
seconds to inflate the balloon.
A. How long would it take to inflate the balloon with 5 quarter-cups of
vinegar? (use technology to graph)
B. What do the x- and y-intercepts of the graph mean in this context?
Do those values make sense?
Try It! Danielle is engineering a new brand of shoes. For x shoes sold, in thousands, a profit of
dollars, in ten thousands, will be earned.
a. How much will be earned in profit for selling 1,000 shoes?
b. What do the x- and y-intercepts mean in this context? Do those values make sense?
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3.1 Additional Practice
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3.2 Adding, Subtracting, and Multiplying Polynomials
Ex 1 How do you add the polynomials? How do you subtract the polynomials?
A. B.
Try It! Add or subtract the polynomials.
a. b.
Ex 2 How do you multiply the polynomials?
A. B.
Try It!
a. b.
Ex 3 Find the area of each figure.
A. B.
Try It! Find the area of each figure.
a. b.
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Ex 4 Carolina makes wind chimes to sell at the local street market. As Carolina produces a greater number of wind
chimes, she can lower the price per unit. The function relates the price v to the number produced
x. The cost c of making x wind chimes can be represented with the function
How many wind chimes should Carolina sell each week to maximize her profit P?
Try It! The function relates the price, v, of Carolinaβs wind chimes to the number produced, x.
The cost of Carolinaβs materials changes so that her new cost function is
Find the new profit function. Then find the quantity that maximizes profit and calculate profit.
Ex 5 Carolinaβs profit function, π¦ = π(π₯) is represented by the graph. Her neighbor Kiyoβs profit from selling x
decorated flower pots can be modeled by the function shown.
A. Find the x- and y-intercepts of each function. Who would lose more money if neither person sold any items?
B. Interpret the end behavior of the functions.
Try It! Compare the profit functions of two additional market sellers modeled by the graph of f and the equation
. Compare and interpret the y-intercepts of these functions and their end behavior.
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3.2 Additional Practice
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3.3 Polynomial Identities
Polynomial Identities
Difference of Squares
Square of a Sum
Difference of Cubes
Sum of Cubes
Ex 1 How can you prove the Sum of Cubes Identity π3 + π3 = (π + π)(π2 β ππ + π2)?
Try It! Prove the Difference of Cubes Identity. π3 β π3 = (π β π)(π2 + ππ + π2)
Ex 2 How can you use polynomial identities to multiply expressions?
A. (2π₯2 + π¦3)2 B. 41 β 39
Try It!
a. b.
Ex 3 How can you use polynomial identities to factor polynomials and simplify numerical expressions?
A. B. C.
Try It! Use polynomial identities to factor each polynomial.
a. b.
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3.3 Additional Practice
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3.4 Dividing Polynomials
Ex 1 How can you use long division to divide P(x) by D(x)? Write the polynomial P(x) in terms of the quotient and
remainder.
A. B.
Try It!
a. b.
Ex 2 What is 2π₯3 β 7π₯2 β 4 divided by π₯ β 3? Try It! Use synthetic division to divide
Use synthetic division. 3π₯3 β 5π₯ + 10 by π₯ β 1.
Ex 3 How is the value of π(π) related to the remainder of π(π₯) Γ· (π₯ β π)?
Try It! Use synthetic division to show that the remainder of π(π₯) = π₯3 + 8π₯2 + 12π₯ + 5 divided by π₯ + 2 is equal
to π(β2).
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The _______________________ states that if a polynomial π(π₯) is divided by π₯ β π, the remainder is π(π).
The _______________________ states that the expression π₯ β π is a factor of a polynomial π(π₯) if and only if π(π) = 0.
Ex 4 The population of tortoises on an island is modeled by the function π(π₯) = βπ₯3 + 6π₯2 + 12π₯ + 325 where x
is the number of years since 2015. Use the Remainder Theorem to estimate the population in 2023.
Try It! A technology company uses the function π (π₯) = βπ₯3 + 12π₯2 + 6π₯ + 80 to model expected annual
revenue, in thousands of dollars, for a new product, where x is the number of years after the product is released. Use
the Remainder Theorem to estimate the revenue in year 5.
Ex 5 How can you use the Remainder and Factor Theorems to determine whether the given binomial is a factor of
P(x)? If it is a factor, write the polynomial in factored form.
A.
B.
Try It! Use the Remainder and Factor Theorems to determine whether the given binomial is a factor of P(x).
a.
b.
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3.4 Additional Practice
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3.5 Zeros of Polynomial Functions
Ex 1 What are the zeros of π(π₯) = π₯(π₯ β 4)(π₯ + 3)? Graph the function.
Try It! Factor the function. Then use the zeros to sketch its graph.
a. b.
Ex 2 How does a multiple zero affect the graph of a polynomial function?
π(π₯) = (π₯ β 1)1(π₯ + 2) π(π₯) = (π₯ β 1)2(π₯ + 2)
π(π₯) = (π₯ β 1)3(π₯ + 2) π(π₯) = (π₯ β 1)4(π₯ + 2)
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Try It! Describe the behavior of the graph of the function at each of its zeros.
a. b.
Ex 3 What are all the real and complex zeros of the polynomial function shown in the graph?
Try It!
a. b.
Ex 4 Acme Innovations makes and sells lamps. Their profit P, in hundreds of dollars earned, is a function of the
number of lamps sold, x in thousands. From historical data, they know that their companyβs profit is modeled by the
function shown. What do the zeros of the function tell you about the number of lamps that Acme Innovations should
produce?
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Try It! Due to a decrease in the cost of materials, the profit function has changed to
How many lamps should they make in order to make a profit?
Ex 5 What are the solutions of 2π₯3 + 5π₯2 β 3π₯ = 3π₯3 + 8π₯2 + 1?
Try It! What is the solution of the equation?
a. b.
Ex 6 What are the solutions of π₯3 β 16π₯ < 0?
Try It! What are the solutions of the inequality?
a. b.
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3.5 Additional Practice
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3.6 Theorems about Roots of Polynomial Equations
Ex 1 From the graph it appears that 4 is a zero of the function π(π₯) = 8π₯5 β 32π₯4 + π₯2 β 4. Without substituting,
how can you determine if 4 is a possible solution to π(π₯) = 0?
Rational Root Theorem
Try It! List all the possible rational solutions for each equation.
a. b.
Ex 2 A storage company is designing a new storage unit. Based on the dimensions shown, the volume of a
container is modeled by the polynomial π£(π₯) = 2π₯3 β 7π₯2 + 6π₯, where x is the width in feet. What are the dimensions
of the container in feet if the volume of the unit is 154 ππ‘3.
Try It! A jewelry box measures 2x + 1 in. long, 2x β 6 in. wide, and x in. tall. The volume of the box is given by the
functionπ£(π₯) = 4π₯3 β 10π₯2 β 6π₯. What is the height of the box, in inches, if its volume is 28 ππ3?
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Fundametal Theorem of Algebra
Ex 3 What are all the complex roots of the polynomial equation? 3π₯4 + 4π₯3 + 2π₯2 β π₯ β 2 = 0
Try It! What are all the complex root of the equation π₯3 β 2π₯2 + 5π₯ β 10 = 0?
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Conjugate Root Theorem
Ex 5
A. What is a quadratic function P with rational coefficients in standard form such that π(π₯) = 0 has 2 + 5π as a root?
B. A polynomial function of degree 4 with rational coefficients has zeros 3 β β7 and 4π. What is a polynomial equation
in standard form with these roots?
Try It!
a. What is a quadratic equation in standard form with rational coefficents that has a root of 5 + 4π?
b. What is a polynomial function Q of degree 4 that has rational coefficients if π(π₯) = 0 has roots 2 β β3 and 5π?
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3.6 Additional Practice
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3.7 Transformations of Polynomial Functions
Odd and Even Functions
A polynomial function is an _________________________ if it is symmetric about the y-axis and an
_________________________ if it symmetric about the origin.
Ex 1 Use the graph to classify the polynomial function.
A. Is it even, odd, or neither? B. Is it even, odd, or neither? C. Is it even, odd, or neither?
Try It! a. Even, odd, or neither? b. Even, odd, or neither?
Ex 2 Is the function odd, even, or neither?
A. B.
Try It! Is the function odd, even, or neither?
a. b.
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Ex 3 How does the transformed graph compare to the graph of the parent function?
A. B.
Try It! How does the graph of the function π(π₯) = 2π₯3 β 5 differ from the graph of its parent function?
Ex 4 The given graph is a transformation of the parent cubic function or parent quartic function. How can you
determine the equation of the graph?
A. B.
Try It! Determine the equation of each graaph as it relates to its parent cubic function or quartic function.
a. b.
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Ex 5
A. The volume of a box, in cubic yards, is given by the function π(π₯) = π₯3. The post office lists permissible shipping
volumes in cubic feet. Write a function with cubic feet as the units.
B. A terrarium is in the shape of a rectangular prism. The volume of the tank is given by π(π₯) = (π₯)(2π₯)(π₯ + 5)
( = 2π₯3 + 10π₯2) where x is measured in inches. The manufacturer wants to compare the volume of this tank with one
that has a width 2 inches shorter but maintains the relationships between the width and the other dimensions. Write a
new function for the volume of this smaller tank.
Try It!
a. The volume of a cube, in cubic feet, is given by the function π(π₯) = π₯3. Write a function for the volume of the cube
with cubic inches as the units.
b. A storage unit is in the shape of a rectangular prism. The volume of the storage unit is given by π(π₯) =
(π₯)(π₯)(π₯ β 1) = π₯3 β π₯2. A potential customer wants to compare the volume of this storage unit with that of another
storage unit that is 1 foot longer in every dimension. Write a function for the volume of this larger unit.
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3.7 Additional Practice
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