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.

Identify and classify polynomials EXAMPLE 2 Identify and classify polynomials 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. 5th degree binomial Yes 7bc3 + 4b4c No; variable exponent n– 2 – 3 6n4 – 8n 2nd degree trinomial 2x2 + x – 5 0 degree monomial 9 Classify by degree and number of terms Is it a polynomial? Expression a. b. c. d. e.

EXAMPLE 3 Add polynomials Find the sum. a. (2x3 – 5x2 + x) + (2x2 + x3 – 1) b. (3x2 + x – 6) + (x2 + 4x + 10)

EXAMPLE 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) = (3x2 + x2) + (x + 4x) + (– 6 + 10) = 4x2 + 5x + 4

EXAMPLE 1 GUIDED PRACTICE Rewrite a polynomial for Examples 1,2, and 3 Write 5y – 2y2 + 9 so that the exponents decrease from left to right. Identify the degree and leading coefficient of the polynomial. 1. – 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

EXAMPLE 3 GUIDED PRACTICE Add polynomials for Examples 1,2, and 3 for Example Find the sum. 3. (5x3 + 4x – 2x) + (4x2 +3x3 – 6) = 8x3 + 4x2 + 2x – 6 ANSWER

EXAMPLE 4 Subtract polynomials Find the difference. a. (4n2 + 5) – (–2n2 + 2n – 4) b. (4x2 – 3x + 5) – (3x2 – x – 8)

EXAMPLE 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 – 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 N and 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 t is 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

EXAMPLE 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.

EXAMPLE 4 GUIDED PRACTICE Subtract polynomials for Examples 4 and 5 Find the difference. 4. a. (4x2 – 7x) – (5x2 + 4x – 9) –x2 – 11x + 9 ANSWER 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

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 No; one exponent is not a whole number. ANSWER 2. – 3b3c4 – 4b2c + c8 ANSWER 8th degree trinomial

Daily Homework Quiz Find the sum or difference. 3. (3m2 – 2m + 9) + (m2 + 2m – 4) 4m2 + 5 ANSWER 4. (– 4a2 + 3a – 1) – (a2 + 2a – 6) ANSWER –5a2 + a + 5

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? about 185 dogs and cats ANSWER