Holt McDougal Algebra 1

6-3 Polynomials

Holt McDougal Algebra 1

6-3 Polynomials

A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents.

Holt McDougal Algebra 1

6-3 Polynomials

Monomials NOT a monomial Reason

5+z A sum is not a monomial

2/n

A monomial cannot

have a variable

denominator

4a

A monomial cannot

have a variable

exponent

x-1

The variable must have

a whole number

exponent.

Holt McDougal Algebra 1

6-3 Polynomials

The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0.

Monomial Degree

10 0

3x 1

1+2=3

-1.8m5 5

Holt McDougal Algebra 1

6-3 Polynomials

Example 1: Finding the Degree of a Monomial

Find the degree of each monomial.

A. 4p4q3

The degree is 7.

B. 7ed

The degree is 2.

C. 3

The degree is 0.

Holt McDougal Algebra 1

6-3 Polynomials

Check It Out! Example 1

Find the degree of each monomial.

a. 1.5k2m

The degree is 3.

b. 4x

The degree is 1.

c. 2c3

The degree is 3.

Holt McDougal Algebra 1

6-3 Polynomials

A polynomial is a monomial or a sum or difference of monomials. Each monomial in a polynomial is called a term. The degree of a polynomial is the degree of the term with the greatest degree.

Holt McDougal Algebra 1

6-3 Polynomials

Polynomials

1252 23 xxxLeading

Coefficient

Degree of

polynomial

Constant

term

Holt McDougal Algebra 1

6-3 Polynomials

Special Polynomials

• Binomial

– Polynomial with two terms

• Trinomial

– Polynomial with three terms

Holt McDougal Algebra 1

6-3 Polynomials

Find the degree of each polynomial.

Example 2: Finding the Degree of a Polynomial

A. 11x7 + 3x3

The degree of the polynomial is the greatest degree, 7.

B.

The degree of the polynomial is the greatest degree, 4.

Holt McDougal Algebra 1

6-3 Polynomials

Check It Out! Example 2

Find the degree of each polynomial.

a. 5x – 6

The degree of the polynomial is the greatest degree, 1.

b. x3y2 + x2y3 – x4 + 2

The degree of the polynomial is the greatest degree, 5.

Holt McDougal Algebra 1

6-3 Polynomials

The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form.

The standard form of a polynomial that contains one variable is written with the terms in order from greatest degree to least degree. When written in standard form, the coefficient of the first term is called the leading coefficient.

Holt McDougal Algebra 1

6-3 Polynomials

Write the polynomial in standard form. Then give the leading coefficient.

Example 3A: Writing Polynomials in Standard Form

6x – 7x5 + 4x2 + 9

–7x5 + 4x2 + 6x + 9. The standard form is The leading

coefficient is –7.

Holt McDougal Algebra 1

6-3 Polynomials

Write the polynomial in standard form. Then give the leading coefficient.

Example 3B: Writing Polynomials in Standard Form

y2 + y6 – 3y

The standard form is The leading coefficient is 1.

y6 + y2 – 3y.

Holt McDougal Algebra 1

6-3 Polynomials

A variable written without a coefficient has a coefficient of 1. Remember “the understood 1”

Remember!

y5 = 1y5

Holt McDougal Algebra 1

6-3 Polynomials

Check It Out! Example 3a

Write the polynomial in standard form. Then give the leading coefficient.

16 – 4x2 + x5 + 9x3

The standard form is The leading

coefficient is 1. x5 + 9x3 – 4x2 + 16.

Holt McDougal Algebra 1

6-3 Polynomials

Check It Out! Example 3b

Write the polynomial in standard form. Then give the leading coefficient.

18y5 – 3y8 + 14y

The standard form is The leading coefficient is –3.

–3y8 + 18y5 + 14y.

Holt McDougal Algebra 1

6-3 Polynomials

Some polynomials have special names based on their degree and the number of terms they have.

Degree Name

0

1

2

Constant

Linear

Quadratic

3

4

5

6 or more 6th,7th,degree and so on

Cubic

Quartic

Quintic

Name Terms

Monomial

Binomial

Trinomial

Polynomial 4 or more

1

2

3

Holt McDougal Algebra 1

6-3 Polynomials

Classify each polynomial according to its degree and number of terms.

Example 4: Classifying Polynomials

A. 5n3 + 4n cubic binomial.

B. 4y6 – 5y3 + 2y – 9 6th-degree polynomial.

C. –2x linear monomial.

Holt McDougal Algebra 1

6-3 Polynomials

Classify each polynomial according to its degree and number of terms.

Check It Out! Example 4

a. x3 + x2 – x + 2 cubic polynomial.

b. 6 constant monomial.

c. –3y8 + 18y5 + 14y 8th-degree trinomial.

Holt McDougal Algebra 1

6-3 Polynomials Example 2

Tell whether is a polynomial. If it is a polynomial, find

its degree and classify it by the number of its terms.

Otherwise, tell why it is not a polynomial.

Quintic binomial Yes

No; negative exponent

No; variable exponent

Quadratic trinomial Yes

constant monomial Yes

7bc3 + 4b4c

n– 2 – 3

6n4 – 8n

2x2 + x – 5

9

Classify by degree and

number of terms

Is it a polynomial? Expression

a.

b.

c.

d.

e.

Holt McDougal Algebra 1

6-3 Polynomials

A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial –16t2 + 220, where t is time in seconds. How far above the water will the lip balm be after 3 seconds?

Example 5: Application Continued

After 3 seconds the lip balm will be 76 feet from the water.

Holt McDougal Algebra 1

6-3 Polynomials

Check It Out! Example 5

What if…? Another firework with a 5-second fuse is launched from the same platform at a speed of 400 feet per second. Its height is given by –16t2 +400t + 6. How high will this firework be when it explodes?

1606 feet

Holt McDougal Algebra 1

6-3 Polynomials Check It Out! Example 4a

Simplify. All variables represent nonnegative numbers.

= xy

Holt McDougal Algebra 1

6-3 Polynomials

Solve for the missing exponent.

16?4 6255 dd

21?56 5000105 aaa