# 1439049084_231925.pdf - Appendix A.3 A27 Polynomials and...

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Appendix A.3 Polynomials and Factoring A27 Polynomials The most common type of algebraic expression is the polynomial. Some examples are and The first two are polynomials in and the third is a polynomial in and The terms of a polynomial in have the form where is the coefficient and is the degree of the term. For instance, the polynomial has coefficients 2, 0, and 1. Polynomials with one, two, and three terms are called monomials, binomials, and trinomials, respectively. In standard form, a polynomial is written with descending powers of Writing Polynomials in Standard Form Leading Polynomial Standard Form Degree Coefficient a. 7 b. 2 c. 8 8 0 8 Now try Exercise 19. A polynomial that has all zero coefficients is called the zero polynomial, denoted by 0. No degree is assigned to this particular polynomial. For polynomials in more than one variable, the degree of a term is the sum of the exponents of the variables in the term. The degree of the polynomial is the highest degree of its terms. For instance, the degree of the polynomial is 11 because the sum of the exponents in the last term is the greatest. The leading coefficient of the polynomial is the coefficient of the highest-degree term. Expressions are not polynomials if a variable is underneath a radical or if a polynomial expression (with degree greater than 0) is in the denominator of a term. The following expressions are not polynomials. The exponent “ ” is not an integer. The exponent “ ” is not a nonnegative integer. 1 x 2 5 x x 2 5 x 1 1 2 x 3 3 x x 3 3 x 1 2 2 x 3 y 6 4 xy x 7 y 4 8 8 x 0 9 9 x 2 4 4 9 x 2 5 5 x 7 4 x 2 3 x 2 4 x 2 5 x 7 2 3 x Example 1 x . 5, 2 x 3 5 x 2 1 2 x 3 5 x 2 0 x 1 k a ax k , x y . x x 5 x 2 y 2 xy 3. 3 x 4 7 x 2 2 x 4, 2 x 5, A.3 P OLYNOMIALS AND F ACTORING What you should learn Write polynomials in standard form. Add, subtract, and multiply polynomials. Use special products to multiply polynomials. Remove common factors from polynomials. Factor special polynomial forms. Factor trinomials as the product of two binomials. Factor polynomials by grouping. Why you should learn it Polynomials can be used to model and solve real-life problems. For instance, in Exercise 224 on page A37, a polynomial is used to model the volume of a shipping box. Definition of a Polynomial in x Let be real numbers and let be a nonnegative integer. A polynomial in is an expression of the form where The polynomial is of degree is the leading coefficient, and is the constant term. a 0 a n n , a n 0. a n x n a n 1 x n 1 . . . a 1 x a 0 x n a 0 , a 1 , a 2 , . . . , a n
A28 Appendix A Review of Fundamental Concepts of Algebra Operations with Polynomials You can add and subtract polynomials in much the same way you add and subtract real numbers. Simply add or subtract the like terms (terms having the same variables to the same powers) by adding their coefficients. For instance, and are like terms and their sum is Sums and Differences of Polynomials a.
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