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Rewrite a polynomial

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EXAMPLE 1

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.

Expression

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

1.

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

3.

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)

EXAMPLE 4

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

EXAMPLE 5

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?

EXAMPLE 5

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.

4.

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

ANSWER

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)

ANSWER

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?