Chapter 5

5.1 Polynomials and Functions

A polynomial function is a function of the form

f (x) = an x n + an – 1 x

n – 1 +· · ·+ a 1 x + a 0

Where an 0 and the exponents are all whole numbers.

A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

For this polynomial function, an is the leading coefficient,

a 0 is the constant term, and n is the degree.

an 0

an

an leading coefficient

a 0

a0 constant term n

n

degree

descending order of exponents from left to right.

n n – 1

Degree Type Standard Form

You are already familiar with some types of polynomialfunctions. Here is a summary of common types ofpolynomial functions.

4 Quartic f (x) = a4 x 4 + a 3 x

3 + a 2 x 2 + a 1 x + a 0

0 Constant f (x) = a 0

3 Cubic f (x) = a 3 x 3 + a 2 x

2 + a 1 x + a 0

2 Quadratic f (x) = a 2 x 2 + a 1 x + a 0

1 Linear f (x) = a1x + a 0

Identifying Polynomial Functions

Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.

f (x) = x 2

– 3x4 – 712

SOLUTION

The function is a polynomial function.

It has degree 4, so it is a quartic function.

The leading coefficient is – 3.

Its standard form is f (x) = – 3x 4

+ x 2 – 7. 1

2

Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.

Identifying Polynomial Functions

The function is not a polynomial function because the

term 3

x does not have a variable base and an exponentthat is a whole number.

SOLUTION

f (x) = x 3 + 3

x

Identifying Polynomial Functions

Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.

SOLUTION

f (x) = 6x 2 + 2 x

–1 + x

The function is not a polynomial function because the term2x

–1 has an exponent that is not a whole number.

Identifying Polynomial Functions

Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.

SOLUTION

The function is a polynomial function.

It has degree 2, so it is a quadratic function.

The leading coefficient is .

Its standard form is f (x) = x2 – 0.5x – 2.

f (x) = – 0.5 x + x 2 – 2

f (x) = x 2 – 3 x

4 – 712

Identifying Polynomial Functions

f (x) = x 3 + 3x

f (x) = 6x2 + 2 x– 1 + x

Polynomial function?

f (x) = – 0.5x + x2 – 2

Decide whether the function is a polynomial function. If it is, write the function in standard form and state the degree and leading coefficient.

Test 1, 2

Goal 1: To evaluate polynomial functions

Goal 2: To simplify polynomial functions

Chapter 5

5.2 Addition and Subtraction of Polynomials

Goal 1: To add polynomial functionsGoal 2: To subtract polynomial functions

The additive inverse of a polynomial.

The additive inverse of a polynomial can be found by replacing each term by its additive inverse.

The sum of a polynomial and its additive inverse is O.

The additive inverse of a polynomial.

The additive inverse of a polynomial can be found by replacing each term by its additive inverse.

The sum of a polynomial and its additive inverse is O.

Thus, to subtract one polynomial from another, we add its additive inverse.

Simplify the polynomial

Test 1, 2

HW #5.1-2Pg 208-209 1-29 Odd, 30-36Pg 212-213 1-31 Odd, 33-35

HW Quiz HW #5.1-2Wednesday, April 19, 2023

Pg 208 32 Pg 208 34 Pg 212 31 Pg 212 34

Pg 208 30 Pg 208 32 Pg 212 34 Pg 212 35

Chapter 5

5.3 Multiplication of Polynomials

Find the product.

Test

Simplify.

Challenge

Based on your answers to parts to the above, write a general formula. Use “2n” to represent a general even integer and let “2n + 1” represent a general odd integer, and use “…” for missing terms.

Answers to challenge

HW #5.3Pg 217-218 1-39 Odd, 40-49

HW Quiz HW #5.3Wednesday, April 19, 2023

Missing Parts

5.4 Factoring

Do Examples from Regular book la205bad

HW 5.4 Pg 222-223 3-60 Every Third, 61-76

5.5 More Factoring

HW Handout Factoring

5.6 Factoring A General Strategy

Do bonus problems from Great Factoring Problems WS

HW Pg 231 1-37 Odd, 38-47

HW Quiz HW #5.6Wednesday, April 19, 2023

Row 1, 3, 5

Factor Completely

1.

2.

3.

4.

Row 2, 4, 6

Factor Completely

1.

2.

3.

4.4 29 81x x

2 6 72x x 2 18 72x x 4 218 81x x

34 108x

4 16x 34 108x

4 64x

5.7 Solving by Factoring

HW #5.7Pg 233 1-42 Left Column, 43-46

Pg 228 89-99 Odd

5.8 Using Polynomial Equations

A candy factory needs a box that has a volume of 30 cubic inches. The width should be 2 inches less than the height and the length should be 5 inches greater than the height. What should the dimensions of the box be?

For the city park commission, you are designing a marble planter in which to plant flowers. You want the length of the planter to be six times the height and the width to be three times the height. The sides should be one foot thick. Since the planter will be on the sidewalk, it does not need a bottom. What should the outer dimensions of the planter be if it is to hold 4 cubic feet of dirt?

Suppose you have 250 cubic inches of clay with which to make a rectangular prism for a sculpture. If you want the height and width each to be 5 inches less than the length, what should the dimensions of the prism be?

HW #5.8a Pg 235-236 1-17 Odd, 18-20

5 15 17 18

7 13 17 20

Test Review

Geometry Express the area A of a rectangle as a function of the length x if the length of the rectangle is twice its width.

Geometry Express the area A of an isosceles right triangle as a function of the length x of one of the two equal sides.

1. Find the degree of a polynomial

2. Find the degree of a monomial

3. Find the function value of a polynomial

4. Standard form of a polynomial

5. Find volume of a box

6. Properties of Exponents

7. Additive Inverse

8. Polynomial arithmetic

9. Factoring

10. Solving Polynomial equations

11. Study all the challenge problems in the book

2 6

6

If and areconstantsand factorsinto

( )( )findk.

h k x kx

x x h

Evaluate: (30)2 + 2(30)(22) + (22)2

2 27 10 15and ( 2 ) 5,

Find ( 5 )

x xy y x y

x y

Factor

1. a9 b9c9d2 - d2

2. x2n + xn

3. yn – yn-1 + yn-2

4. 72x2n + 120xn + 50

5. 50x4 – 72y6

6. x2 – w2 –16 + 8w

Factor

1. x4 + 9x2 + 81

2. x4 + x2y2 + 25y4

3. x4 + 64

4. 7y2a + b – 5ya + b + 3ya + 2b

5. 4x2a – 4xa – 3

6. a6 – 64b6

7. 9a3 +9a2b – 4ab2 – 4b3

HW #R-4Pg 241 1-42