chapter 5 polynomials, polynomial functions, and factoring

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Chapter 5 Polynomials, Polynomial Functions, and Factoring

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Chapter 5 Polynomials, Polynomial Functions, and Factoring. § 5.1. Introduction to Polynomials and Polynomial Functions. Polynomials. A polynomial is a single term or the sum of two or more terms containing variables with whole number exponents. . Terms. Consider the polynomial:. - PowerPoint PPT Presentation

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Page 1: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Chapter 5Polynomials, Polynomial Functions, and Factoring

Page 2: Chapter 5 Polynomials, Polynomial Functions, and Factoring

§ 5.1

Introduction to Polynomials and Polynomial Functions

Page 3: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 5.1

Polynomials

A polynomial is a single term or the sum of two or more terms containing variables with whole number exponents.

Consider the polynomial: 6523 34 xxxThis polynomial contains four terms. It is customary to write the terms inorder of descending powers of the variable. This is the standard form of apolynomial.

55 9 2

2

1 54 , or , 7 , 6 , , and 9.2 2 3

mx m z x zx

Terms

Page 4: Chapter 5 Polynomials, Polynomial Functions, and Factoring

3 2 2 33 5, 4 5 8, and 5x m m p t s

1 2 13 , 9 , and x x xx

• Polynomials

• Not Polynomials

A polynomial is a term or a finite sum of terms in which all variables have whole number exponents and no variables appear in

denominators.

Polynomials

Page 5: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 5.1

Polynomials

The degree of a polynomial is the greatest degree of any term of the polynomial. The degree of a term

6523 34 xxx

mn yax is (n +m)

and the coefficient of the term is a. If there is exactly one term of greatest degree, it is called the leading term. It’ s coefficient is called the leading coefficient. Consider the polynomial:

3 is the leading coefficient. The degree is 4.

Page 6: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 5.1

PolynomialsCONTINUE

D

The degree of the polynomial is the greatest degree of all its terms, which is 10. The leading term is the term of the greatest degree, which is . Its coefficient, -5, is the leading coefficient.

735 yx

Page 7: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 5.1

PolynomialsEXAMPLE

SOLUTION

Determine the coefficient of each term, the degree of each term, the degree of the polynomial, the leading term, and the leading coefficient of the polynomial.

735 yx

Term Coefficient Degree (Sum of Exponents on the Variables)

12 4 + 1 = 5-5 3 + 7 = 10-1 2 + 0 = 2

4 4 0 + 0 = 0

4512 2734 xyxyx

2x

yx412

Page 8: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 5.1

Polynomials

2524)( 23 xxxxf

is an example of a polynomial function. In a polynomialfunction, the expression that defines the function is a polynomial.

How do you evaluate a polynomial function? Use substitution just as you did to evaluate functions in Chapter 2.

Page 9: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 5.1

PolynomialsEXAMPLE

SOLUTION

The polynomial function

models the cumulative number of deaths from AIDS in the United States, f (x), x years after 1990. Use this function to solve the following problem.

Find and interpret f (8).

896,107575,572212 2 xxxf

To find f (8), we replace each occurrence of x in the function’s formula with 8.

Page 10: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 5.1

Polynomials

896,107575,572212 2 xxxf

896,1078575,57822128 2 f

896,1078575,576422128 f

Thus, f (8) = 426,928. According to this model, this means that 8 years after 1990, in 1998, there had been 426,928 cumulative deaths from AIDS in the United States.

CONTINUED

896,107600,460568,1418 f

928,4268 f

Original functionReplace each occurrenceof x with 8Evaluate exponentsMultiplyAdd

Page 11: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 4.1

PolynomialsCheck Point

2

2 find

6534

function polynomial For the23

fxxxxf

16

62523242 23 f

6101232

Page 12: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 5.1

Polynomials

Polynomial functions of degree 2 or higher have graphs that are smooth and continuous.

By smooth, we mean that the graph contains only rounded corners with no sharp corners.

By continuous, we mean that the graph has no breaks and can be drawn without lifting the pencil from the rectangular coordinate system.

Page 13: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 5.1

Graphs of PolynomialsEXAMPLE

The graph below does not represent a polynomial function. Although it has a couple of smooth, rounded corners, it also has a sharp corner and a break in the graph. Either one of these last two features disqualifies it from being a polynomial function.

Smooth rounded curve

Smooth rounded curve

Discontinuous break

Sharp Corner

Page 14: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 5.1

Adding Polynomials

EXAMPLE

SOLUTIONAdd: . 1771119131167 2323 xxxxxx

1771119131167 2323 xxxxxx

1771119131167 2323 xxxxxx Remove parentheses

4 4 5 12

1713711116197

23

2233

xxx

xxxxxx Rearrange terms so that like terms are adjacent

Combine like terms

Polynomials are added by removing the parentheses that surround each polynomial (if any) and then combining like terms.

Page 15: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 4.1

Adding Polynomials p 311

Check Point 6

1364

34723

23

xx

xx

10103 23 xx

Page 16: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #16 Section 4.1

Adding PolynomialsCheck Point

7

91282

357 23

23

yxyxy

yxyxy

91539 23 yxyxy

Page 17: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #17 Section 5.1

Subtracting Polynomials

EXAMPLE

SOLUTIONSubtract . 8653765 324324 xyyxyxyyxyx

xyyxyxyyxyx 8653765 324324

xyyxyxyyxyx 8653765 324324 Change subtraction to addition and change the sign of every term of the polynomial in parentheses.

Rearrange terms

Combine like terms

xyyxyx

xyyyxyxyxyx

8 11 2

8675635

324

332424

To subtract two polynomials, change the sign of every term of the second polynomial. Add this result to the first polynomial.

Page 18: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #18 Section 4.1

Subtracting PolynomialsCheck Point

8

)173(4 -

)95(14 23

23

xxx

xxx

108210 23 xxx

1734-

9514 23

23

xxx

xxx

Page 19: Chapter 5 Polynomials, Polynomial Functions, and Factoring

DONE

Page 20: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #20 Section 5.1

Graphs of Polynomials

The Leading Coefficient TestAs x increases or decreases without bound, the graph of a polynomial function eventually rises or falls. In particular,

Odd-Degree Polynomials

If the leading coefficient is positive, the graph falls to the left and rises to the right.

If the leading coefficient is negative, the graph rises to the left and falls to the right.

Even-Degree Polynomials

If the leading coefficient is positive, the graph rises to the left and to the right.

If the leading coefficient is negative, the graph falls to the left and to the right.

Page 21: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #21 Section 5.1

PolynomialsEXAMPLE

SOLUTION

The common cold is caused by a rhinovirus. After x days of invasion by the viral particles, the number of particles in our bodies, f (x), in billions, can be modeled by the polynomial function

Use the Leading Coefficient Test to determine the graph’s end behavior to the right. What does this mean about the number of viral particles in our bodies over time?

.5375.0 34 xxxf

Since the polynomial function has even degree and has a negative leading coefficient, the graph falls to the right (and the left). This means that the viral particles eventually decrease as the days increase.

Page 22: Chapter 5 Polynomials, Polynomial Functions, and Factoring

Blitzer, Intermediate Algebra, 5e – Slide #22 Section 5.1

Polynomials

The Degree of TIf , the degree of is n. The degree of a nonzero constant is 0. The constant 0 has no defined degree.

nax0a

nax

Adding PolynomialsPolynomials are added by removing the parentheses that surround each polynomial (if any) and then combining like terms.

Subtracting PolynomialsTo subtract two polynomials, change the sign of every term of the second polynomial. Add this result to the first polynomial.