Adding and Subtracting Polynomials Lesson 8 1 LEARNING

Adding and Subtracting Polynomials Lesson 8 -1

LEARNING GOAL Understand how to write polynomials in standard form and add and subtract polynomials.

4 x³y²z + 3 xy 4 x² + 3 x + 5

Identify Polynomials State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

A. State whether 3 x 2 + 2 y + z is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. A. yes, monomial B. yes, binomial C. yes, trinomial D. not a polynomial

B. State whether 4 a 2 – b– 2 is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. A. yes, monomial B. yes, binomial C. yes, trinomial D. not a polynomial

C. State whether 8 r – 5 s is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. A. yes, monomial B. yes, binomial C. yes, trinomial D. not a polynomial

D. State whether 3 y 5 is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. A. yes, monomial B. yes, binomial C. yes, trinomial D. not a polynomial

Standard Form of a Polynomial A. Write 9 x 2 + 3 x 6 – 4 x in standard form. Identify the leading coefficient. Step 1 Find the degree of each term. Degree: 2 6 1 Polynomial: 9 x 2 + 3 x 6 – 4 x Step 2 Write the terms in descending order. Answer: 3 x 6 + 9 x 2 – 4 x; the leading coefficient is 3.

Standard Form of a Polynomial B. Write 12 + 5 y + 6 xy + 8 xy 2 in standard form. Identify the leading coefficient. Step 1 Find the degree of each term. Degree: 0 1 2 3 Polynomial: 12 + 5 y + 6 xy + 8 xy 2 Step 2 Write the terms in descending order. Answer: 8 xy 2 + 6 xy + 5 y + 12; the leading coefficient is 8.

A. Write – 34 x + 9 x 4 + 3 x 7 – 4 x 2 in standard form. A. 3 x 7 + 9 x 4 – 4 x 2 – 34 x B. 9 x 4 + 3 x 7 – 4 x 2 – 34 x C. – 4 x 2 + 9 x 4 + 3 x 7 – 34 x D. 3 x 7 – 4 x 2 + 9 x 4 – 34 x

B. Identify the leading coefficient of 5 m + 21 – 6 mn + 8 mn 3 – 72 n 3 when it is written in standard form. A. – 72 B. 8 C. – 6 D. 72

Add Polynomials A. Find (7 y 2 + 2 y – 3) + (2 – 4 y + 5 y 2). Horizontal Method (7 y 2 + 2 y – 3) + (2 – 4 y + 5 y 2) = (7 y 2 + 5 y 2) + [2 y + (– 4 y)] + [(– 3) + 2] Group like terms. = 12 y 2 – 2 y – 1 Combine like terms.

Add Polynomials Vertical Method 7 y 2 + 2 y – 3 (+) 5 y 2 – 4 y + 2 Notice that terms are in descending order with like terms aligned. 12 y 2 – 2 y – 1 Answer: 12 y 2 – 2 y – 1

Add Polynomials B. Find (4 x 2 – 2 x + 7) + (3 x – 7 x 2 – 9). Horizontal Method (4 x 2 – 2 x + 7) + (3 x – 7 x 2 – 9) = [4 x 2 + (– 7 x 2)] + [(– 2 x) + 3 x] + [7 + (– 9)] Group like terms. = – 3 x 2 + x – 2 Combine like terms.

Add Polynomials Vertical Method 4 x 2 – 2 x + 7 2 (+) – 7 x + 3 x – 9 Align and combine like terms. – 3 x 2 + x – 2 Answer: – 3 x 2 + x – 2

A. Find (3 x 2 + 2 x – 1) + (– 5 x 2 + 3 x + 4). A. – 2 x 2 + 5 x + 3 B. 8 x 2 + 6 x – 4 C. 2 x 2 + 5 x + 4 D. – 15 x 2 + 6 x – 4

B. Find (4 x 3 + 2 x 2 – x + 2) + (3 x 2 + 4 x – 8). A. 5 x 2 + 3 x – 6 B. 4 x 3 + 5 x 2 + 3 x – 6 C. 7 x 3 + 5 x 2 + 3 x – 6 D. 7 x 3 + 6 x 2 + 3 x – 6

Subtract Polynomials A. Find (6 y 2 + 8 y 4 – 5 y) – (9 y 4 – 7 y + 2 y 2). Horizontal Method Subtract 9 y 4 – 7 y + 2 y 2 by adding its additive inverse. (6 y 2 + 8 y 4 – 5 y) – (9 y 4 – 7 y + 2 y 2) = (6 y 2 + 8 y 4 – 5 y) + (– 9 y 4 + 7 y – 2 y 2) = [8 y 4 + (– 9 y 4)] + [6 y 2 + (– 2 y 2)] + (– 5 y + 7 y) = –y 4 + 4 y 2 + 2 y

Subtract Polynomials Vertical Method Align like terms in columns and subtract by adding the additive inverse. 8 y 4 + 6 y 2 – 5 y (–) 9 y 4 + 2 y 2 – 7 y 8 y 4 + 6 y 2 – 5 y Add the opposite. (+) – 9 y 4 – 2 y 2 + 7 y –y 4 + 4 y 2 + 2 y Answer: –y 4 + 4 y 2 + 2 y

Subtract Polynomials Find (6 n 2 + 11 n 3 + 2 n) – (4 n – 3 + 5 n 2). Horizontal Method Subtract 4 n 4 – 3 + 5 n 2 by adding the additive inverse. (6 n 2 + 11 n 3 + 2 n) – (4 n – 3 + 5 n 2) = (6 n 2 + 11 n 3 + 2 n) + (– 4 n + 3 – 5 n 2 ) = 11 n 3 + [6 n 2 + (– 5 n 2)] + [2 n + (– 4 n)] + 3 = 11 n 3 + n 2 – 2 n + 3 Answer: 11 n 3 + n 2 – 2 n + 3

Subtract Polynomials Vertical Method Align like terms in columns and subtract by adding the additive inverse. 11 n 3 + 6 n 2 + 2 n + 0 (–) 0 n 3 + 5 n 2 + 4 n – 3 11 n 3 + 6 n 2 + 2 n + 0 Add the opposite. (+) 0 n 3 – 5 n 2 – 4 n + 3 11 n 3 + n 2 – 2 n + 3 Answer: 11 n 3 + n 2 – 2 n + 3

A. Find (3 x 3 + 2 x 2 – x 4) – (x 2 + 5 x 3 – 2 x 4). A. 2 x 2 + 7 x 3 – 3 x 4 B. x 4 – 2 x 3 + x 2 C. x 2 + 8 x 3 – 3 x 4 D. 3 x 4 + 2 x 3 + x 2

B. Find (8 y 4 + 3 y 2 – 2) – (6 y 4 + 5 y 3 + 9). A. 2 y 4 – 2 y 2 – 11 B. 2 y 4 + 5 y 3 + 3 y 2 – 11 C. 2 y 4 – 5 y 3 + 3 y 2 – 11 D. 2 y 4 – 5 y 3 + 3 y 2 + 7

Add and Subtract Polynomials A. VIDEO GAMES The total amount of toy sales T (in billions of dollars) consists of two groups: sales of video games V and sales of traditional toys R. In recent years, the sales of traditional toys and total sales could be modeled by the following equations, where n is the number of years since 2000. R = 0. 46 n 3 – 1. 9 n 2 + 3 n + 19 T = 0. 45 n 3 – 1. 85 n 2 + 4. 4 n + 22. 6 A. Write an equation that represents the sales of video games V.

Add and Subtract Polynomials Find an equation that models the sales of video games V. video games + traditional toys = total toy sales V+R=T V=T–R Subtract the polynomial for R from the polynomial for T. 0. 45 n 3 – 1. 85 n 2 + 4. 4 n + 22. 6 (–) 0. 46 n 3 – 1. 9 n 2 + 3 n + 19

Add and Subtract Polynomials Add the opposite. 0. 45 n 3 – 1. 85 n 2 + 4. 4 n + 22. 6 (+) – 0. 46 n 3 + 1. 9 n 2 – 3 n – 19 – 0. 01 n 3 + 0. 05 n 2 + 1. 4 n + 3. 6 Answer: V = – 0. 01 n 3 + 0. 05 n 2 + 1. 4 n + 3. 6

Add and Subtract Polynomials B. Use the equation to predict the amount of video game sales in the year 2009. The year 2009 is 2009 – 2000 or 9 years after the year 2000. Substitute 9 for n. V = – 0. 01(9)3 + 0. 05(9)2 + 1. 4(9) + 3. 6 = – 7. 29 + 4. 05 + 12. 6 + 3. 6 = 12. 96 Answer: The amount of video game sales in 2009 will be 12. 96 billion dollars.

A. BUSINESS The profit a business makes is found by subtracting the cost to produce an item C from the amount earned in sales S. The cost to produce and the sales amount could be modeled by the following equations, where x is the number of items produced. C = 100 x 2 + 500 x – 300 S = 150 x 2 + 450 x + 200 Find an equation that models the profit. A. 50 x 2 – 50 x + 500 B. – 50 x 2 – 50 x + 500 C. 250 x 2 + 950 x + 500 D. 50 x 2 + 950 x + 100

B. Use the equation 50 x 2 – 50 x + 500 to predict the profit if 30 items are produced and sold. A. $500 B. $30 C. $254, 000 D. $44, 000

Homework p. 469 #21 -49 odd, #55 -69 odd