name: chapter 5: polynomials and polynomial functions

Name: Chapter 5: Polynomials and Polynomial Functions

Lesson 5-1: Solving Polynomial Equations Date:

*Note: A monomial is a number, variable, or an expression that is the product of one or more variables

with nonnegative integer exponents.

*Note: A polynomial is the addition or subtraction of monomials.

* The degree of a polynomial is the degree of the monomial with the highest degree.


Example 1: Simplify each expression. Assume that no variable equals 0.

A. (𝑎−3)(𝑎2𝑏4)(𝑐−1). B. 𝑛2

𝑛10

C. (3𝑎3

𝑏4)

2

D. (𝑥−2𝑦−3)(𝑥−3𝑦5)(𝑧2)

Example 2: Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the

polynomial.

A. 𝑐4 − 4√𝑐 + 18 B. −16𝑝5 +3

4𝑝2𝑞7

C. 𝑥2 − 3𝑥−1 + 7 D. 1

2𝑎2𝑏3 + 3𝑐5

Example 3: Simplify each expression.

A. (2𝑎3 + 5𝑎 − 7) − (𝑎3 − 3𝑎 + 2) B. (4𝑥2 − 9𝑥 + 3) + (−2𝑥2 − 5𝑥 − 6)


Example 4: Simplify −𝑦(4𝑦2 + 2𝑦 − 3)

Example 5: A small online retailer estimates that the cost, in dollars, associated with selling 𝑥 units of a

particular product is given by the expression 0.001𝑥2 + 5𝑥 + 500. The revenue from selling 𝑥 units is

given by 10𝑥. Write a polynomial to represent the profits generated by the product if profit = revenue –

cost.

Example 6: Simplify (𝑎2 + 3𝑎 − 4)(𝑎 + 2)


Lesson 5-2: Dividing Polynomials Date:

Example 1: Simplify 5𝑎2𝑏−15𝑎𝑏3+10𝑎3𝑏4

5𝑎𝑏.

Example 2: Use long division to find (𝑥2 − 2𝑥 − 15) ÷ (𝑥 − 5)

Example 3: MULTIPLE CHOICE Which expression is equal to (𝑎2 − 5𝑎 + 3)(2 − 𝑎)−1?

A. 𝑎 + 3

B. −𝑎 + 3 +3

2−𝑎

C. −𝑎 − 3 +3

2−𝑎

D. −𝑎 + 3 −3

2−𝑎


Synthetic division is a simpler process for dividing a polynomial by a binomial.

Example 4: Use synthetic division to find

A. (𝑥3 − 4𝑥2 + 6𝑥 − 4) ÷ (𝑥 − 2) B. (𝑥2 + 8𝑥 + 7) ÷ (𝑥 + 1)

Example 5: Use synthetic division to find

A. (4𝑦3 − 6𝑦2 + 4𝑦 − 1) ÷ (2𝑦 − 1) B. (8𝑦3 − 12𝑦2 + 4𝑦 + 10) ÷ (2𝑦 + 1)


Lesson 5-3: Polynomial Functions Date:

A polynomial is written in standard form when the values of the exponents are in descending order.

The coefficient of the first term of a polynomial in standard form is called the leading coefficient.

Polynomial Expression Degree Leading Coefficient

Constant

Linear

Quadratic

Cubic

General

Example 1: State the degree and leading coefficient of each polynomial in one variable. If it is not a

polynomial in one variable, explain why.

A. 7𝑧3 − 4𝑧2 + 𝑧 B. 6𝑎3 − 4𝑎2 + 𝑎𝑏2

C. 3𝑥5 + 2𝑥2 − 4 − 8𝑥6 D. 9𝑦3 + 4𝑦6 − 45 − 8𝑦2 − 5𝑦7

A polynomial function is a continuous function that can be described by a polynomial equation in one

variable. The simplest polynomial functions of the form 𝑓(𝑥) = 𝑎𝑥𝑏 are called power functions.

Example 2: The volume of air in the lungs during a 5-second respiratory cycle can be modeled by 𝑣(𝑡) =

– 0.037𝑡3 + 0.152𝑡2 + 0.173𝑡, where 𝑣 is the volume in liters and 𝑡 is the time in seconds. This model is

an example of a polynomial function. Find the volume of air in the lungs 1.5 seconds into the respiratory

cycle.


Example 3: Find 𝑏(2𝑥 − 1) − 3𝑏(𝑥) if 𝑏(𝑚) = 2𝑚2 + 𝑚 − 1.


Constant

Linear

Quadratic

Cubic

Quartic

Quintic


Example 4: For each graph describe the end behavior, determine whether it represents an odd-degree or an

even-degree function, and state the number of real zeros.

A. B.

C. D.


Lesson 5-4: Analyzing Graphs of Polynomial Functions Date:

Example 1: Graph 𝑓(𝑥) = −𝑥3 − 4𝑥2 + 5 by making a table of values.

Example 2: Determine consecutive values of 𝑥 between which each real zero of the function 𝑓(𝑥) = 𝑥4 −

𝑥3 − 4𝑥2 + 1 is located. Then draw the graph.

x

y

x

y


If no other points nearby have a greater y-coordinate, it is a relative maximum.

If no other points nearby have a lesser y-coordinate, it is a relative minimum.

The maximum and minimum values of a function are called the extrema or turning points.

Example 3: Graph 𝑓(𝑥) = 𝑥3 − 3𝑥2 + 5. Estimate the x-coordinates at which the relative maxima and

relative minima occur.

Example 4: The weight 𝑤, in pounds, of a patient during a 7-week illness is modeled by the function

𝑤(𝑛) = 0.1𝑛3– 0.6𝑛2 + 110, where 𝑛 is the number of weeks since the patient became ill.

A. Graph the equation.

B. Describe the turning points of the graph and its end behavior.

C. What trends in the patient’s weight does the graph suggest?

D. Is it reasonable to assume the trend will continue indefinitely?

x

y

x

y


Lesson 5-5: Solving Polynomial Equations Date:

Polynomials that cannot be factored are called prime polynomials.

Example 1: Factor each polynomial. If the polynomial cannot be factored, write prime.

A. 𝑥3 − 400 B. 25𝑥5 + 3𝑥2𝑦3



A. 𝑥3 + 5𝑥2 − 2𝑥 − 10 B. 𝑎2 + 3𝑎𝑦 + 2𝑎𝑦2 + 6𝑦3


A. 𝑥2𝑦3 − 3𝑥𝑦3 + 2𝑦3 + 𝑥2𝑧3 − 3𝑥𝑧3 + 2𝑧3 B. 64𝑥6 − 𝑦6

Example 4: Determine the dimensions of the cubes below if the length of the smaller cube is one half the

length of the larger cube, and the volume of the shaded figure is 23,625 cubic centimeters.


Example 5: Write each expression in quadratic form, if possible.

A. 2𝑥6 − 𝑥3 + 9 B. 𝑥4 − 2𝑥3 − 1

Example 6: Solve 𝑥4 − 29𝑥2 + 100 = 0.


Lesson 5-6: The Remainder and Factor Theorems Date:

The value of 𝑓(𝑟) is the same as the remainder when the polynomial is divided by 𝑥 − 𝑟.

Applying the Remainder Theorem using synthetic division to evaluate a function is called synthetic

substitution.

Example 1: If 𝑓(𝑥) = 2𝑥4 − 5𝑥2 + 8𝑥 − 7, find 𝑓(6).

Example 2: The number of college students from the United States who study abroad can be modeled by

the function 𝑆(𝑥) = 0.02𝑥4 − 0.52𝑥3 + 4.03𝑥2 + 0.09𝑥 + 77.54, where 𝑥 is the number of years since

1993 and 𝑆(𝑥) is the number of students in thousands. How many U.S. college students will study abroad in

2011?


When you divide a polynomial by one of its binomial factors, the quotient is called a depressed

polynomial, which has a degree that is one less than the original polynomial.

Example 3:

A. Determine whether 𝑥 − 3 is a factor of 𝑥3 + 4𝑥2 − 15𝑥 − 18. If so, find the remaining factors of the

polynomial.

B. Determine whether 𝑥 + 2 is a factor of 𝑥3 + 8𝑥2 + 17𝑥 + 10. If so, find the remaining factors of the

polynomial.


Lesson 5-7: Roots and Zeros Date:

Example 1: Solve each equation. State the number and type of roots.

A. 𝑥2 + 2𝑥 − 48 = 0 B. 𝑦4 − 256 = 0


Example 2: State the possible number of positive real zeros, negative real zeros, and imaginary zeros of

𝑝(𝑥) = −𝑥6 + 4𝑥3 − 2𝑥2 − 𝑥 − 1.

Example 3: Find all of the zeros of 𝑓(𝑥) = 𝑥3 − 𝑥2 + 2𝑥 + 4.


Example 4: Write a polynomial function of least degree with integral coefficients, the zeros of which include

4 and 4 – 𝑖.


Lesson 5-8: Rational Zero Theorem Date:

Example 1: List all of the possible rational zeros of each function.

A. 𝑓(𝑥) = 3𝑥4 − 𝑥3 + 4 B. 𝑓(𝑥) = 𝑥4 + 7𝑥3 − 15

Example 2: The volume of a rectangular solid is 1120 cubic feet. The width is 2 feet less than the height, and

the length is 4 feet more than the height. Find the dimensions of the solid.

Example 3: Find all of the zeros of 𝑓(𝑥) = 𝑥4 + 𝑥3 − 19𝑥2 + 11𝑥 + 30.

name: chapter 5: polynomials and polynomial functions

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