SECTION A.3 Polynomials A27 5 Factor Polynomials Consider the following product: x x x x 2 3 4 2 5 12 2 ( )( ) + − = − − The two polynomials on the left side are called factors of the polynomial on the right side. Expressing a given polynomial as a product of other polynomials—that is, finding the factors of a polynomial—is called factoring . We restrict our discussion here to factoring polynomials in one variable into products of polynomials in one variable, where all coefficients are integers. We call this factoring over the integers . Any polynomial can be written as the product of 1 times itself or as 1− times its additive inverse. If a polynomial cannot be written as the product of two other polynomials (excluding 1 and 1− ), then the polynomial is prime . When a polynomial has been written as a product consisting only of prime factors, it is factored completely . Examples of prime polynomials (over the integers) are x x x x x 2, 3, 5, , 1, 1, 3 4, 4 2 + − + + Dividing Two Polynomials Find the quotient and the remainder when x x x x x 3 2 5 is divided by 1 4 3 2 − + − − + EXAMPLE 7 The process of dividing two polynomials leads to the following result: Solution In setting up this division problem, it is necessary to leave a space for the missing x2 term in the dividend. ) − − − + − + − − + − − + − − + − − + − − + − − x x x x x x x x x x x x x x x x x x x x x 2 3 1 3 2 5 2 2 5 2 2 2 3 4 5 3 3 3 2 2 2 4 3 4 3 2 3 2 3 2 2 2 Check: Quotient Divisor Remainder ⋅ + x x x x x x x x x x x x x x x x x 2 3 1 2 2 2 2 3 3 3 2 3 2 5 Dividend 2 2 4 3 2 3 2 2 4 3 ( )( ) = − − − + + − =−+− + −− +−+− = − + − = As a result, x x x x x x x x x x 3 2 5 1 2 3 2 1 4 3 2 2 2 − + − − + = − − + − − + Divisor → Subtract → Subtract → Subtract → ← Quotient ← Dividend ← Remainder Now Work problem 65 THEOREM Let Q be a polynomial of positive degree, and let P be a polynomial whose degree is greater than or equal to the degree of Q .The remainder after dividing P by Q is either the zero polynomial or a polynomial whose degree is less than the degree of the divisor Q. COMMENT Over the real numbers, x3 4 + factors into ( ) +x 3 . 4 3 It is the fraction 4 3 that causes +x3 4 to be prime over the integers. ■
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