13 Exponents and Polynomials

13.1 Adding and Subtracting Polynomials Objectives 1. Review combining like terms. 2. Know the vocabulary for polynomials. 3. Evaluate polynomials. 4. Add polynomials. 5. Subtract polynomials. 6. Add and subtract polynomials with more than one variable.

Review Combining Like Terms Example 1 Simplify each expression by adding like terms. (a) –3 x + 7 x 5 = (–3 + 7) x 5 Distributive property = 4 x 5 (b) 2 n + 5 n – n 3 = (2 + 5 – 1) n 3 Distributive property = 6 n 3

Review Combining Like Terms Example 1 (concluded) Simplify each expression by adding like terms. (c) 4 g h – 9 g h + 2 g h 7 4 = (4 – 9 + 2) g h 7 4 Distributive property = – 3 g h 7 4 4

Know the Vocabulary for Polynomials Example 2 Simplify each polynomial if possible. Then give the degree and tell whether the polynomial is a monomial, a binomial, a trinomial, or none of these. (a) 2 t + 7 4 The polynomial cannot be simplified. The degree is 4. Two terms. The polynomial is a binomial. (b) 3 e + 5 e – 9 e 2 The polynomial can be simplified. The degree is 2. = – e 2 The simplified polynomial is a monomial. One term.

Evaluate a Polynomial Example 3 (a) Find the value of 5h4 – 3h2 + 4h + 7 when h = –3. Substitute h = –3. Apply the exponents. Multiply. Add and subtract.

(b) Find the value of 5h4 – 3h2 + 4h + 7 when h = 2. Evaluate a Polynomial Example 3(concluded) (b) Find the value of 5h4 – 3h2 + 4h + 7 when h = 2. Substitute h = 2. Apply the exponents. Multiply. Add and subtract. 7

Add Polynomials Adding Polynomials To add two polynomials, add like terms.

Write like terms in columns. Add Polynomials Example 4 8y – 7y – y + 3 3 2 Write like terms in columns. 6y + 2y – 4y + 1 3 2 14y 3 2 – 5y – 5y + 4 Now add, column by column.

Write like terms in columns. Add Polynomials Example 4 (concluded) 7n 3 + 2n Write like terms in columns. n – 9n + 12 2 7n 3 2 + n – 7n + 12 Now add, column by column.

Add Polynomials Example 5 ( 2n – 7n – 4 ) + ( – 5n – 8n + 10 ) – 3n = + 6

Add Polynomials Example 5 (concluded) ( 7p – 9p – 4 ) + ( 5p – 1 ) = 3 ( 7p – 9p – 4 ) + ( 5p – 1 ) 3 = 7p – 4p – 5

Subtract Polynomials Subtracting Polynomials To subtract two polynomials, change all the signs of the second polynomial and add the result to the minuend (first polynomial).

Subtract Polynomials Example 6 ( 3x – 5 ) – ( 6x – 4 ) = ( 3x – 5 ) + ( – 6x + 4 ) = – 3x – 1 Change the signs in the second polynomial and add.

Change the signs in the second polynomial and add. Subtract Polynomials Example 6 (concluded) ( 7y + 8 ) – ( y + 4y – 2 ) 3 2 Write the problem. = ( 7y + 8 ) + ( – y – 4y + 2 ) 3 2 6y 3 – 4y 2 = + 10 Change the signs in the second polynomial and add.

Arrange like terms in columns. Subtract Polynomials Example 7 4g + 6g – g – 5 4 3 4g + 6g – g – 5 4 3 – 6g + 2g – 4g + 3 4 3 6g – 2g + 4g – 3 4 3 10g 4 + 4g 3 + 3g – 8 Arrange like terms in columns. Change all signs in the second row, then add.

Example 8 Perform the indicated operations to simplify the expression. Subtract Polynomials Example 8 Perform the indicated operations to simplify the expression. ( 2 – 3m + 5m ) – ( 6 + 4m – 7m ) + ( 9 – m + 3m ). 2 Rewrite, changing the subtraction to adding the opposite. ( 2 – 3m + 5m ) + ( – 6 – 4m + 7m ) + ( 9 – m + 3m ) 2 = Combine like terms. 2 + 15m = 5 – 8m

Add and Subtract Multivariable Polynomials Example 9 (a) Add or subtract as indicated. ( 6c – 2cd + d ) + ( 3c + 9cd – 4d ) = 9c + 7cd – 3d

Add and Subtract Multivariable Polynomials Example 9 (concluded) (b) Add or subtract as indicated. ( 2a b – 4ab + b ) – ( 5a b – 3ab + 7b ) 2 2 2 2 2 = 2a b – 4ab + b – 5a b + 3ab – 7b = – 3a b 2 – 6b 2 – ab