51 R.5 Polynomials Concept Check Answer each question. 117. If the lengths of the sides of a cube are tripled, by what factor will the volume change? x x x 3x 3x 3x 118. If the radius of a circle is doubled, by what factor will the area change? r 2r R.5 Polynomials ■ Polynomials ■ Addition and Subtraction ■ Multiplication ■ Division Polynomials The product of a number and one or more variables raised to powers is a term. The number is the numerical coefficient, or just the coefficient, of the variables. The coefficient of the variable in -3m4 is -3, and the coefficient in -p2 is -1. Like terms are terms with the same variables each raised to the same powers. -13x3, 4x3, -x3 Like terms 6y, 6y2, 4y3 Unlike terms A polynomial is a term or a finite sum of terms, with only whole number exponents permitted on the variables. If the terms of a polynomial contain only the variable x, then the polynomial is a polynomial in x. 5x3 - 8x2 + 7x - 4, 9p5 - 3, 8r2, 6 Polynomials Some algebraic expressions are not polynomials. x-1 + 3x-2, 29 - x, 1 x Not polynomials The first has negative integer exponents, the second involves a variable under a radical, and the third has a variable in the denominator. Some polynomials with a specific number of terms are given special names. • A polynomial with exactly three terms is a trinomial. • A two-term polynomial is a binomial. • A single-term polynomial is a monomial. The degree of a term with one variable is the exponent on the variable. For example, the degree of 2x3 is 3, and the degree of 17x (that is, 17x1) is 1. Although many polynomials contain only one variable, polynomials may have more than one variable. The degree of a term with more than one variable is the sum of the exponents on the variables. The degree of a polynomial is the greatest degree of all of its terms.
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