Polynomials Copyright Cengage Learning All rights reserved

Polynomials Copyright © Cengage Learning. All rights reserved. 4

Section 4. 5 Adding and Subtracting Polynomials Copyright © Cengage Learning. All rights reserved.

Objectives 1 Add two or more monomials. 2 Subtract two monomials. 3 Add two polynomials. 4 Subtract two polynomials. 5 Simplify an expression using the order of operations and combining like terms. 6 Solve an application requiring operations with polynomials. 3

1. Add two or more monomials 4

Add two or more monomials Recall Unlike terms • 3 xyz 2 • – 2 xyz 2 Unlike terms • ab 2 c • 5 a 2 bd 2 Also recall that to combine like terms, we add (or subtract) their coefficients and keep the same variables with the same exponents. 5

Add two or more monomials For example, 2 y + 5 y = (2 + 5)y = 7 y Likewise, 4 x 3 y 2 + 9 x 3 y 2 = 13 x 3 y 2 – 3 x 2 + 7 x 2 = (– 3 + 7)x 2 = 4 x 2 4 r 2 s 3 t 4 + 7 r 2 s 3 t 4 = 11 r 2 s 3 t 4 6

Example Perform the following additions. a. 5 xy 3 + 7 xy 3 = 12 xy 3 b. – 7 x 2 y 2 + 6 x 2 y 2 + 3 x 2 y 2 = –x 2 y 2 + 3 x 2 y 2 = 2 x 2 y 2 c. (2 x 2)2 + 81 x 4 = 4 x 4 + 81 x 4 = 85 x 4 (2 x 2)2 = (2 x 2) = 4 x 4 7

2. Subtract two monomials 8

Subtract two monomials To subtract one monomial from another, we add the opposite of the monomial that is to be subtracted. In symbols, x – y = x + (–y). 9

Example Find each difference. a. 8 x 2 – 3 x 2 = 8 x 2 + (– 3 x 2) = 5 x 2 b. 6 x 3 y 2 – 9 x 3 y 2 = 6 x 3 y 2 + (– 9 x 3 y 2) = – 3 x 3 y 2 c. – 3 r 2 st 3 – 5 r 2 st 3 = – 3 r 2 st 3 + (– 5 r 2 st 3) = – 8 r 2 st 3 10

3. Add two polynomials 11

Add two polynomials Because of the distributive property, we can remove parentheses enclosing several terms when the sign preceding the parentheses is +. +(3 x 2 + 3 x – 2) = +1(3 x 2 + 3 x – 2) = 1(3 x 2) + 1(3 x) + 1(– 2) = 3 x 2 + 3 x + (– 2) = 3 x 2 + 3 x – 2 We can add polynomials by removing parentheses, if necessary, and then combining any like terms that are contained within the polynomials. 12

Example Add: (3 x 2 – 3 x + 2) + (2 x 2 + 7 x – 4) Solution: (3 x 2 – 3 x + 2) + (2 x 2 + 7 x – 4) = 3 x 2 – 3 x + 2 x 2 + 7 x – 4 Remove parentheses. = 3 x 2 + 2 x 2 – 3 x + 7 x + 2 – 4 Use the commutative property of addition. = 5 x 2 + 4 x – 2 Combine like terms. 13

Add two polynomials Additions such as Example 3 often are written with like terms aligned vertically. We then can add the polynomials column by column. 14

4. Subtract two polynomials 15

Subtract two polynomials We can remove parentheses enclosing several terms when the sign preceding the parentheses is negative by distributing a – 1 to each term within the parentheses. –(3 x 2 + 3 x – 2) = – 1(3 x 2) + (– 1)(3 x) + (– 1)(– 2) = – 3 x 2 + (– 3 x) + 2 = – 3 x 2 – 3 x + 2 This suggests that the way to subtract polynomials is to remove parentheses by changing the sign of all terms being subtracted and then combine like terms. 16

Example Subtract: a. (3 x – 4) – (5 x + 7) = 3 x – 4 – 5 x – 7 = – 2 x – 11 b. (3 x 2 – 4 x – 6) – (2 x 2 – 6 x + 12) = 3 x 2 – 4 x – 6 – 2 x 2 + 6 x – 12 = x 2 + 2 x – 18 c. (– 4 rt 3 + 2 r 2 t 2) – (– 3 rt 3 + 2 r 2 t 2) = – 4 r t 3 + 2 r 2 t 2 + 3 r t 3 – 2 r 2 t 2 = –r t 3 17

Subtract two polynomials To subtract polynomials in vertical form, we add the negative of the subtrahend (the bottom polynomial) to the minuend (the top polynomial) to obtain the difference. 18

5. Simplify an expression using the order of operations and combining like terms 19

Simplify an expression using the order of operations and combining like terms Because of the distributive property, we can remove parentheses enclosing several terms that are multiplied by a monomial by multiplying every term within the parentheses by that monomial. For example, to add 3(2 x + 5) and 2(4 x – 3), we proceed as follows: 3(2 x + 5) + 2(4 x – 3) = 6 x + 15 + 8 x – 6 = 6 x + 8 x + 15 – 6 Use the commutative property of addition. = 14 x + 9 Combine like terms. 20

Example Simplify: a. 3(x 2 + 4 x) + 2(x 2 – 4) = 3 x 2 + 12 x + 2 x 2 – 8 = 5 x 2 + 12 x – 8 b. 8(y 2 – 2 y + 3) – 4(2 y 2 + y – 3) = 8 y 2 – 16 y + 24 – 8 y 2 – 4 y + 12 = – 20 y + 36 c. – 4(x 2 y 2 – x 2 y + 3 x) – (x 2 y 2 – 2 x) + 3(x 2 y 2 + 2 x 2 y) = – 4 x 2 y 2 + 4 x 2 y – 12 x – x 2 y 2 + 2 x + 3 x 2 y 2 + 6 x 2 y = – 2 x 2 y 2 + 10 x 2 y – 10 x 21

6. Solve an application requiring operations with polynomials 22

Example 9 – Property Values A house purchased for $95, 000 is expected to appreciate according to the formula y 1 = 2, 500 x + 95, 000, where y 1 is the value of the house after x years. A second house purchased for $125, 000 is expected to appreciate according to the formula y 2 = 4, 500 x + 125, 000. Find one formula that will give the value of both properties after x years. Solution: The value of the first house after x years is given by the polynomial 2, 500 x + 95, 000. 23

Example 9 – Solution cont’d The value of the second house after x years is given by the polynomial 4, 500 x + 125, 000. The value of both houses will be the sum of these two polynomials. 2, 500 x + 95, 000 + 4, 500 x + 125, 000 = 7, 000 x + 220, 000 The total value y of the properties is given by y = 7, 000 x + 220, 000. 24