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Rewrite a polynomial

Write 15x – x3 + 3 so that the exponents decrease from left to right. Identify the degree and leading coefficient of the polynomial.

SOLUTION

Consider the degree of each of the polynomial’s terms.

15x – x3 + 3

The polynomial can be written as – x3 +15 + 3. The greatest degree is 3, so the degree of the polynomial is 3, and the leading coefficient is –1.

Is it a polynomial?

Classify by degree and number of terms

a.

9

Yes

0 degree monomial

b.

2x2 + x – 5

Yes

2nd degree trinomial

c.

6n4 – 8n

No; variable exponent

d.

n– 2 – 3

No; variable exponent

e.

7bc3 + 4b4c

Yes

5th degree binomial

EXAMPLE 2

Identify and classify polynomials

Tell whetheris 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.

EXAMPLE 3

Add polynomials

Find the sum.

a. (2x3 – 5x2 + x) + (2x2 + x3 – 1)

b. (3x2 + x – 6) + (x2 + 4x + 10)

+ x3 + 2x2 – 1

EXAMPLE 3

Add polynomials

SOLUTION

a. Vertical format: Align like terms in vertical columns.

(2x3 – 5x2 + x)

3x3 – 3x2 + x – 1

b. Horizontal format: Group like terms and simplify.

(3x2 + x – 6) + (x2 + 4x + 10) =

(3x2+ x2) + (x+ 4x) + (– 6+ 10)

= 4x2 + 5x + 4

Write 5y – 2y2 + 9 so that the exponents decrease from left to right. Identify the degree and leading coefficient of the polynomial.

2.

Tell whether y3 – 4y + 3 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.

ANSWER

– 2y2 +5y + 9 Degree: 2, Leading Coefficient: –2

ANSWER

polynomial Degree: 3, trinomial

EXAMPLE 1

for Examples 1,2, and 3

Rewrite a polynomial

GUIDED PRACTICE

Find the sum.

ANSWER

= 8x3 + 4x2+ 2x – 6

EXAMPLE 3

for Example

for Examples 1,2, and 3

Add polynomials

GUIDED PRACTICE

(5x3 + 4x – 2x) + (4x2 +3x3 – 6)

Subtract polynomials

Find the difference.

a. (4n2 + 5) – (–2n2 + 2n – 4)

b. (4x2 – 3x + 5) – (3x2 – x – 8)

–(–2n2 + 2n – 4)

2n2 – 2n + 4

EXAMPLE 4

Subtract polynomials

SOLUTION

a. (4n2 + 5)

4n2 + 5

6n2 – 2n + 9

b. (4x2 – 3x + 5) – (3x2 – x – 8) =

4x2 – 3x + 5– 3x2 + x + 8

= (4x2– 3x2) +(–3x+x) + (5+ 8)

=x2–2x+13

Solve a multi-step problem

BASEBALL ATTENDANCE

Major League Baseball teams are divided into two leagues. During the period 1995–2001, the attendance Nand A (in thousands) at National and American League baseball games, respectively, can be modeled by

N = –488t2 + 5430t + 24,700 and

A = –318t2 + 3040t + 25,600

where tis the number of years since 1995. About how many people attended Major League Baseball games in 2001?

Solve a multi-step problem

SOLUTION

STEP 1

Add the models for the attendance in each league to find a model for M, the total attendance (in thousands).

M =(–488t2 + 5430t + 24,700) +(–318t2 + 3040t + 25,600)

= (–488t2– 318t2) + (5430t+ 3040t) + (24,700 + 25,600)

= –806t2 + 8470t + 50,300

M = –806(6)2 + 8470(6) + 50,300 72,100

ANSWER

About 72,100,000 people attended Major League

Baseball games in 2001.

EXAMPLE 5

Solve a multi-step problem

STEP 2

Substitute 6 for tin the model, because 2001 is 6 years

after 1995.

Find the difference.

BASEBALL ATTENDNCE Look back at Example 5. Find the difference in attendance at National and American League baseball games in 2001.

5.

ANSWER

–x2 – 11x + 9

ANSWER

about 7,320,000 people

EXAMPLE 4

for Examples 4 and 5

Subtract polynomials

GUIDED PRACTICE

a. (4x2 – 7x) – (5x2 + 4x – 9)

No; one exponent is not a whole number.

ANSWER

ANSWER

8th degree trinomial

Daily Homework Quiz

If the expression is a polynomial, find its degree and classify it by the number of terms. Otherwise, tell why it is not a polynomial.

1. m3 + n4m2 + m–2

2. – 3b3c4 – 4b2c + c8

4m2 + 5

ANSWER

–5a2 + a + 5

Daily Homework Quiz

Find the sum or difference.

3. (3m2 – 2m + 9) + (m2 + 2m– 4)

4. (– 4a2 + 3a – 1) – (a2 + 2a – 6)

about 185 dogs and cats

Daily Homework Quiz

5. The number of dog adoptions D and cat adoptions C can be modeled by D = 1.35 t2 – 9.8t + 131 and

C= 0.1t2 – 3t + 79 where t represents the years since 1998. About how many dogs and cats were adopted in 2004?

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