rewrite a polynomial

Warm-Up ExercisesRewrite a polynomialEXAMPLE 1

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.

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.

15x – x3 + 3

Warm-Up Exercises

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.

EXAMPLE 2 Identify and classify polynomials

5th degree binomialYes7bc3 + 4b4c

No; variable exponentn– 2 – 3

No; variable exponent6n4 – 8n

2nd degree trinomialYes2x2 + x – 5

0 degree monomialYes9

Classify by degree and number of terms

Is it a polynomial?Expression

a.

b.c.d.e.

Warm-Up ExercisesEXAMPLE 3 Add polynomials

Find the sum.

a. (2x3 – 5x2 + x) + (2x2 + x3 – 1)

b. (3x2 + x – 6) + (x2 + 4x + 10)

Warm-Up ExercisesEXAMPLE 3 Add polynomials

SOLUTION

a. Vertical format: Align like terms in vertical columns. (2x3 – 5x2 + x)

+ x3 + 2x2 – 1

3x3 – 3x2 + x – 1

b. Horizontal format: Group like terms and simplify.

(3x2 + x – 6) + (x2 + 4x + 10) =

= 4x2 + 5x + 4

(3x2 + x2) + (x + 4x) + (– 6 + 10)

Warm-Up ExercisesRewrite a polynomialEXAMPLE 1

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

1.

GUIDED PRACTICE for Examples 1,2, and 3

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

ANSWER

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.

2.

ANSWER

polynomial Degree: 3, trinomial

Warm-Up ExercisesEXAMPLE 3 Add polynomials

(5x3 + 4x – 2x) + (4x2 +3x3 – 6)

Find the sum.3.

GUIDED PRACTICE for Example for Examples 1,2, and 3

= 8x3 + 4x2 + 2x – 6ANSWER

Warm-Up ExercisesEXAMPLE 4 Subtract polynomials

Find the difference.

a. (4n2 + 5) – (–2n2 + 2n – 4)

b. (4x2 – 3x + 5) – (3x2 – x – 8)

Warm-Up ExercisesEXAMPLE 4 Subtract polynomials

SOLUTION

a. (4n2 + 5) 4n2 + 5

–(–2n2 + 2n – 4) 2n2 – 2n + 4

6n2 – 2n + 9

b. (4x2 – 3x + 5) – (3x2 – x – 8) =

= (4x2 – 3x2) + (–3x + x) + (5 + 8)

= x2 – 2x + 13

4x2 – 3x + 5 – 3x2 + x + 8

Warm-Up ExercisesEXAMPLE 5 Solve a multi-step problem

BASEBALL ATTENDANCE

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

N = –488t2 + 5430t + 24,700 and

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

A = –318t2 + 3040t + 25,600

Warm-Up ExercisesEXAMPLE 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

Warm-Up ExercisesEXAMPLE 5 Solve a multi-step problem

STEP 2

Substitute 6 for t in the model, because 2001 is 6 years after 1995.

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

ANSWER

About 72,100,000 people attended Major League Baseball games in 2001.

Warm-Up ExercisesEXAMPLE 4 Subtract polynomials

a. (4x2 – 7x) – (5x2 + 4x – 9)

GUIDED PRACTICE for Examples 4 and 5

Find the difference.4.

–x2 – 11x + 9ANSWER

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

5.

ANSWER about 7,320,000 people

Warm-Up ExercisesDaily 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

No; one exponent is not a whole number.

ANSWER

2. – 3b3c4 – 4b2c + c8

ANSWER 8th degree trinomial

Warm-Up ExercisesDaily Homework Quiz

Find the sum or difference.

3. (3m2 – 2m + 9) + (m2 + 2m – 4)

4m2 + 5ANSWER

4. (– 4a2 + 3a – 1) – (a2 + 2a – 6)

ANSWER –5a2 + a + 5

Warm-Up ExercisesDaily 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?

about 185 dogs and catsANSWER