R 3 Addition Subtraction and Multiplication of Polynomials

R. 3 Addition, Subtraction, and Multiplication of Polynomials · · Identify the terms, coefficients, and the degree of a polynomial. Add, subtract, and multiply polynomials. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Polynomials are a type of algebraic expression. Examples: 5 y 6 t Copyright ©

Polynomials in One Variable A polynomial in one variable is any expression of the type where n is a nonnegative integer, an, …, a 0 are real numbers, called coefficients. The parts of the polynomial separated by plus signs are called terms. The leading coefficient is an, and the constant term is a 0. If an 0, the degree of the polynomial is n. The polynomial is said to be written in descending order, because the exponents decrease from left to right. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 3 - 3

Examples Identify the terms of the polynomial. 4 x 7 3 x 5 + 2 x 2 9 The terms are: 4 x 7, 3 x 5, 2 x 2, and 9. Find the degree of each polynomial. Polynomial Degree a) 7 x 5 3 5 b) x 2 + 3 x + 4 x 3 3 c) 5 0 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 3 - 4

Degree of a Polynomial An algebraic expression like 5 a 3 b + 2 ab – 1 is a polynomial in several variables. The degree of a term is the sum of the exponents of the variables in that term. The degree of a polynomial is the degree of the term of the highest degree. The degrees of the terms of 5 a 3 b + 2 ab – 1 are 4, 2, and 0. The degree of the polynomial is 4. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 3 - 5

Addition and Subtraction If two terms of an expression have the same variables raised to the same powers, they are called like terms, or similar terms. Like Terms 3 y 2 + 7 y 2 4 x 3 2 x 3 Unlike Terms 9 w 3 y We add or subtract polynomials by combining like terms. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 3 - 6

Examples Add: ( 4 x 4 + 3 x 2 x) + (3 x 4 5 x 2 + 7) = ( 4 x 4 + 3 x 4) + (3 x 2 5 x 2) x + 7 = ( 4 + 3)x 4 + (3 5)x 2 x + 7 = x 4 2 x 2 x + 7 Subtract: 8 x 3 y 2 5 xy (4 x 3 y 2 + 2 xy) = 8 x 3 y 2 5 xy 4 x 3 y 2 2 xy = 4 x 3 y 2 7 xy Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 3 - 7

Multiplication is based on the distributive property. Example Multiply: (5 x 1)(2 x + 5) = 5 x(2 x + 5) – 1(2 x + 5) = 10 x 2 + 25 x 2 x 5 = 10 x 2 + 23 x 5 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 3 - 8

Multiplication To multiply two polynomials in general, we multiply each term of one by each term of the other and add the products. Example: (3 x 3 y 5 x 2 y + 5 y)(4 y 6 x 2 y) 3 x 3 y(4 y 6 x 2 y) 5 x 2 y(4 y 6 x 2 y) + 5 y(4 y 6 x 2 y) = 12 x 3 y 2 18 x 5 y 2 20 x 2 y 2 + 30 x 4 y 2 + 20 y 2 30 x 2 y 2 = 12 x 3 y 2 18 x 5 y 2 20 x 2 y 2 30 x 2 y 2 + 30 x 4 y 2 + 20 y 2 = 18 x 5 y 2 + 30 x 4 y 2 + 12 x 3 y 2 50 x 2 y 2 + 20 y 2 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 3 - 9

Multiplication - FOIL We can find the product of two binomials by multiplying the First terms, then the Outer terms, then the Inner terms, then the Last terms. Then we combine like terms if possible. This procedure is called FOIL. Example: Multiply (x 5)(4 x 1) F O I L (x 5)(4 x 1) = 4 x 2 – x – 20 x + 5 = 4 x 2 – 21 x + 5 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 3 - 10

Multiplication Special Products of Binomials (A + B)2 = (A+B) Square of a sum = A 2 + AB + B 2 = A 2 + 2 • AB + B 2 (A – B)2 = (A – B) Square of a difference = A 2 – AB + B 2 = A 2 – 2 • AB + B 2 (A + B)(A – B) = A 2 – AB +AB – B 2 Product of a sum = A 2 – B 2 and a difference Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 3 - 11

Special Product Examples Multiply: (6 x 1)2 = (6 x)2 – 2 • 6 x + 12 = 36 x 2 12 x + 1 Multiply: (2 x 3)(2 x + 3) = (2 x)2 – 32 = 4 x 2 9 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide R. 3 - 12