52 CHAPTER R Review of Basic Concepts EXAMPLE 1 Classifying Expressions as Polynomials Identify each as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these. (a) 9x7 - 4x3 + 8x2 (b) 2t4 - 1 t2 (c) - 4 5 x3y2 SOLUTION (a) 9x7 - 4x3 + 8x2 is a polynomial. The first term, 9x7, has greatest degree, so this is a polynomial of degree 7. Because it has three terms, it is a trinomial. (b) 2t4 - 1 t2 is not a polynomial because it has a variable in the denominator. (c) -4 5x 3y2 is a polynomial. Add the exponents 3 + 2 = 5 to determine that it is of degree 5. Because there is one term, it is a monomial. S Now Try Exercises 11, 13, and 19. Addition and Subtraction Polynomials are added by adding coefficients of like terms. They are subtracted by subtracting coefficients of like terms. y = 1y EXAMPLE 2 Adding and Subtracting Polynomials Add or subtract, as indicated. (a) 12y4 - 3y2 + y2 + 14y4 + 7y2 + 6y2 (b) 1-3m3 - 8m2 + 42 - 1m3 + 7m2 - 32 (c) 18m4p5 - 9m3p52 + 111m4p5 + 15m3p52 (d) 41x2 - 3x + 72 - 512x2 - 8x - 42 SOLUTION (a) 12y4 - 3y2 + y2 + 14y4 + 7y2 + 6y2 = 12 + 42y4 + 1-3 + 72y2 + 11 + 62y Add coefficients of like terms. = 6y4 + 4y2 + 7y Work inside the parentheses. (b) 1-3m3 - 8m2 + 42 - 1m3 + 7m2 - 32 = 1-3 - 12m3 + 1-8 - 72m2 + 34 - 1-324 = -4m3 - 15m2 + 7 (c) 18m4p5 - 9m3p52 + 111m4p5 + 15m3p52 = 19m4p5 + 6m3p5 Polynomials and Their Degrees Type of Polynomial Example Degree Monomial 7 0 17 = 7x02 5x3y7 10 13 + 7 = 102 Binomial 6 + 2x3 3 11y + 8 1 1y = y12 Trinomial t2 + 11t + 4 2 -3 + 2k5 + 9z4 5 x3y9 + 12xy4 + 7xy 12 (The terms have degrees 12, 5, and 2, and 12 is the greatest.) Subtract coefficients of like terms. Simplify.
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