rewrite a polynomial

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Warm-Up Exercises Rewrite a polynomial EXAMPLE 1 Write 15x x 3 + 3 so that the exponents decrease from left to right. Identify the degree and leading coefficient of the polynomial. SOLUTION nsider the degree of each of the polynomial’s terms The polynomial can be written as x 3 +15 + 3. The greatest degree is 3, so the degree of the polynomial is 3, and the leading coefficient is –1. 15x x 3 + 3

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EXAMPLE 1. Rewrite a polynomial. Write 15 x – x 3 + 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. 15 x – x 3 + 3. - PowerPoint PPT Presentation

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Page 1: 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

Page 2: Rewrite a polynomial

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.

Page 3: Rewrite a polynomial

Warm-Up ExercisesEXAMPLE 3 Add polynomials

Find the sum.

a. (2x3 – 5x2 + x) + (2x2 + x3 – 1)

b. (3x2 + x – 6) + (x2 + 4x + 10)

Page 4: Rewrite a polynomial

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)

Page 5: Rewrite a polynomial

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

Page 6: Rewrite a polynomial

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

Page 7: Rewrite a polynomial

Warm-Up ExercisesEXAMPLE 4 Subtract polynomials

Find the difference.

a. (4n2 + 5) – (–2n2 + 2n – 4)

b. (4x2 – 3x + 5) – (3x2 – x – 8)

Page 8: Rewrite a polynomial

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

Page 9: Rewrite a polynomial

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

Page 10: Rewrite a polynomial

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

Page 11: Rewrite a polynomial

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.

Page 12: Rewrite a polynomial

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

Page 13: Rewrite a polynomial

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

Page 14: Rewrite a polynomial

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

Page 15: Rewrite a polynomial

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