CHAPTER 4 Polynomials: Operations Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 4.1Integers as Exponents 4.2Exponents and Scientific Notation 4.3Introduction to Polynomials 4.4Addition and Subtraction of Polynomials 4.5Multiplication of Polynomials 4.6Special Products 4.7Operations with Polynomials in Several Variables 4.8Division of Polynomials

OBJECTIVES 4.7 Operations with Polynomials in Several Variables Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. aEvaluate a polynomial in several variables for given values of the variables. bIdentify the coefficients and the degrees of the terms of a polynomial and the degree of a polynomial. cCollect like terms of a polynomial. dAdd polynomials. eSubtract polynomials. fMultiply polynomials.

EXAMPLE Solution We substitute  3 for x and 4 for y: 5 + 4x + xy 2 + 9x 3 y 2 = 5 + 4(  3) + (  3)(4 2 ) + 9(  3) 3 (4) 2 = 5  12  48  3888 =  Operations with Polynomials in Several Variables a Evaluate a polynomial in several variables for given values of the variables. AEvaluate the polynomial 5 + 4x + xy 2 + 9x 3 y 2 for x =  3 and y = 4. Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE The surface area of a right circular cylinder is given by the polynomial 2  rh + 2  r 2 where h is the height and r is the radius of the base. A barn silo has a height of 50 feet and a radius of 9 feet. Approximate its surface area. Solution We evaluate the polynomial for h = 50 ft and r = 9 ft. If 3.14 is used to approximate , we have h r 4.7 Operations with Polynomials in Several Variables a Evaluate a polynomial in several variables for given values of the variables. BApplications of Polynomials Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE h = 50 ft and r = 9 ft 4.7 Operations with Polynomials in Several Variables a Evaluate a polynomial in several variables for given values of the variables. AApplications of Polynomials Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 2  rh + 2  r 2  2(3.14)(9 ft)(50 ft) + 2(3.14)(9 ft) 2  2(3.14)(9 ft)(50 ft) + 2(3.14)(81 ft 2 )  2826 ft ft 2  ft 2 Note that the unit in the answer (square feet) is a unit of area. The surface area is about ft 2 (square feet).

Recall that the degree of a monomial is the number of variable factors in the term. 4.7 Operations with Polynomials in Several Variables b Identify the coefficients and the degrees of the terms of a polynomial and the degree of a polynomial. Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

Example Identify the coefficient and the degree of each term and the degree of the polynomial: 10x 3 y 2 – 15xy 3 z 4 + yz + 5y + 3x Operations with Polynomials in Several Variables b Identify the coefficients and the degrees of the terms of a polynomial and the degree of a polynomial. Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

TermCoefficientDegreeDegree of the Polynomial 10x 3 y 2  15xy 3 z 4 yz 5y5y 3x23x2 9 10x 3 y 2 – 15xy 3 z 4 + yz + 5y + 3x Operations with Polynomials in Several Variables b Identify the coefficients and the degrees of the terms of a polynomial and the degree of a polynomial. Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

Like, or similar terms either have exactly the same variables with exactly the same exponents or are constants. For example, 9w 5 y 4 and 15w 5 y 4 are like terms and –12 and 14 are like terms, but –6x 2 y and 9xy 3 are not like terms. 4.7 Operations with Polynomials in Several Variables c Collect like terms of a polynomial. Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a) 10x 2 y + 4xy 3  6x 2 y  2xy 3 = (10  6)x 2 y + (4  2)xy 3 = 4x 2 y + 2xy Operations with Polynomials in Several Variables c Collect like terms of a polynomial. CCombine like terms. Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a) 10x 2 y + 4xy 3  6x 2 y  2xy Operations with Polynomials in Several Variables c Collect like terms of a polynomial. CCombine like terms. Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE b) 8st  6st 2 + 4st 2 + 7s st  12s 3 + t  Operations with Polynomials in Several Variables c Collect like terms of a polynomial. CCombine like terms. Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution (–6x 3 + 4y – 6y 2 ) + (7x 3 + 5x 2 + 8y 2 ) = x 3 + 5x 2 + 4y + 2y Operations with Polynomials in Several Variables d Add polynomials. DAdd: (–6x 3 + 4y – 6y 2 ) + (7x 3 + 5x 2 + 8y 2 ) Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE (5x 2 y + 2x 3 y 2 + 4x 2 y 3 + 7y)  (5x 2 y  7x 3 y 2 + x 2 y 2  6y) Solution (5x 2 y + 2x 3 y 2 + 4x 2 y 3 + 7y)  (5x 2 y  7x 3 y 2 + x 2 y 2  6y) = = 9x 3 y 2 + 4x 2 y 3  x 2 y y 4.7 Operations with Polynomials in Several Variables e Subtract polynomials. ESubtract: Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 4.7 Operations with Polynomials in Several Variables f Multiply polynomials. F Multiply: (4x 2 y  3xy + 4y)(xy + 3y) Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Solution 4x 2 y  3xy + 4y xy + 3y 12x 2 y 2  9xy y 2 4x 3 y 2  3x 2 y 2 + 4xy 2 4x 3 y 2 + 9x 2 y 2  5xy y 2

EXAMPLE 4.7 Operations with Polynomials in Several Variables f Multiply polynomials. FMultiply: Slide 17Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. (4x 2 y  3xy + 4y)(xy + 3y)

4.7 Operations with Polynomials in Several Variables f Multiply polynomials. Slide 18Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. The special products discussed in Section 4.5 can speed up your work.

EXAMPLE 4.7 Operations with Polynomials in Several Variables f Multiply polynomials. GMultiply. (continued) Slide 19Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. a) (x + 6y)(2x  3y) Solution = 2x 2  3xy + 12xy  18y 2 = 2x 2 + 9xy  18y 2 FOIL

EXAMPLE b) (5x + 7y) 2 = c) (a 4  5a 2 b 2 ) 2 = 4.7 Operations with Polynomials in Several Variables f Multiply polynomials. GMultiply. (continued) Slide 20Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE d) (7a 2 b + 3b)(7a 2 b  3b) = 4.7 Operations with Polynomials in Several Variables f Multiply polynomials. GMultiply. (continued) Slide 21Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE e) (  3x 3 y 2 + 7t)(3x 3 y 2 + 7t) = (7t  3x 3 y 2 )(7t + 3x 3 y 2 ) = (7t) 2  (3x 3 y 2 ) 2 = 49t 2  9x 6 y Operations with Polynomials in Several Variables f Multiply polynomials. GMultiply. Slide 22Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE f) (3x + 1  4y)(3x y) ((3x+1) – 4y)((3x+1) + 4y) = (3x + 1) 2  (4y) 2 = 9x 2 + 6x + 1  16y Operations with Polynomials in Several Variables f Multiply polynomials. GMultiply. Slide 23Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.