Chapter 6

Exponents and Polynomials

§ 6.1

Adding and Subtracting Polynomials

Blitzer, Introductory Algebra, 5e – Slide #3 Section 6.1

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 xxx

This polynomial contains four terms. It is customary to write the terms in order of descending powers of the variable. This is the standard form of a polynomial. Here are two other polynomials which are written in standard form.

638

82754

23

xx

xxx

Blitzer, Introductory Algebra, 5e – Slide #4 Section 6.1

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.

The Degree of axn • If a ≠ 0, the degree of axn is n. The degree of a nonzero constant

term is 0. The constant 0 has no defined degree.

8275 23 xxx

Degree 3 Degree 2 Degree 1

Degree of nonzero constant: 0

Polynomials

Blitzer, Introductory Algebra, 5e – Slide #5 Section 6.1

• The degree of a polynomial is the degree of its highest order term.

• Example:

Degree 3 Polynomial:

Degree 4 Polynomial: 638

82754

23

xx

xxx

Degree of a Polynomial

Blitzer, Introductory Algebra, 5e – Slide #6 Section 6.1

• Monomial: A polynomial with one term.• Binomial: A polynomial with two terms.• Trinomial: A polynomial with three terms.

Example: This is a 4th degree trinomial.

638 4 xx

Special Polynomials

Blitzer, Introductory Algebra, 5e – Slide #7 Section 6.1

Blitzer, Introductory Algebra, 5e – Slide #8 Section 6.1

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.

• Polynomials are added by combining like terms. • Like terms are terms containing exactly the same

variables to the same powers.

• Example:2222 10)64(64 xxxx

These like terms both contain x to the second power.

Add the coefficients and keep the same variable

factor, x2

Adding Polynomials

Blitzer, Introductory Algebra, 5e – Slide #9 Section 6.1

518123

)5()18(123

)72()810()48()52(

)7845()21082 :Add

23

23

2233

2323

xxx

xxx

xxxxxx

xxxx-xx(-

Group like terms.

Combine like terms.

Adding Polynomials

Blitzer, Introductory Algebra, 5e – Slide #10 Section 6.1

EXAMPLEEXAMPLE

Blitzer, Introductory Algebra, 5e – Slide #11 Section 6.1

Adding Polynomials

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Add: . 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

943x

9)4(3

)72()610()58(

) 765()2108(

)765()2108 :Subtract

2

2

22

22

22

x

xx

xxxx

xxxx

xxxx(

Add the opposite of the polynomial being subtracted.

Group like terms.

Combine like terms.

Subtracting Polynomials

Blitzer, Introductory Algebra, 5e – Slide #12 Section 6.1

EXAMPLEEXAMPLE

Blitzer, Introductory Algebra, 5e – Slide #13 Section 6.1

Subtracting Polynomials

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Subtract . 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

Blitzer, Introductory Algebra, 5e – Slide #14 Section 6.1

Polynomial Functions

2524)( 23 xxxxf

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

How do you evaluate a polynomial function? Use Substitution.

Blitzer, Introductory Algebra, 5e – Slide #15 Section 6.1

Graphs of Polynomial Functions

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 page.

Blitzer, Introductory Algebra, 5e – Slide #16 Section 6.1

Graphs of Polynomials

EXAMPLEEXAMPLE

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