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Math 102 4.2 "Remainder and Factor Theorems" Objectives: * Use the remainder theorem to evaluate a function for a given value. * Determine if an expression is a factor of a given polynomial. * Find linear factors of a polynomial. Remainder Theorem Let’s consider the division algorithm when the dividend, f (x), is divided by a linear polynomial of the form x the division algorithm f (x) = g (x) q (x) + r (x) and r (x) is the remainder) becomes f (x) = (x than the degree of the divisor, x f (x) = (x c) q (x) + R : ; c. Then (where f (x) is the dividend, g (x) is the divisor, q (x) is the quotient, c) q (x) + r (x) : Because the degree of the remainder, r (x) ; must be less c; the remainder is a constant. Therefore, if we let R represent the remainder, we have If we evaluate f at c, we obtain : In other words, if a polynomial is divided by a linear polynomial of the form x c, then the remainder is the value of the polynomial at c. Remainder Theorem: kIf a polynomial f (x) is divided by x c, then the remainder is equal to f (c) :k Example 1: (Using the remainder theorem) Find f (c) (i) by using synthetic division and the remainder theorem and (ii) by evaluating f (c) directly. 3 4 a) f (x) = x + x 2 2x 4 and c = 1 b) f (x) = 2x + x3 4x2 x + 1 and c = 2 Factor Theorem A general factor theorem can be formulated by considering the equation f (x) = (x of f (x), then the remainder R must be zero. Conversely, if R = f (c) = 0; then f (x) = (x c) q (x) + R : If x c is a factor c) q (x) : In other words, x c is a factor of f (x). Factor Theorem: kA polynomial f (x) has a factor x c if and only if f (c) = 0:k Page: 1 Notes by Bibiana Lopez College Algebra by Kaufmann and Schwitters 4.2 Example 2: (Using the factor theorem) Use the factor theorem to help answer each question about factors. a) Is x + 3 a factor of 6x2 + 13x 15? b) Is x 1 a factor of 3x3 + 5x2 x 2? Example 3: (Using the factor theorem) Use synthetic division to show that g (x) is a factor of f (x) ; and complete the factorization of f (x) : a) g (x) = x 1; f (x) = 3x3 + 19x2 b) g (x) = x + 2; f (x) = x3 + 7x2 + 4x 38x + 16 12 Example 4: (Using the factor theorem) Find the values of k that make x 1 a factor of k 2 x4 + 3kx2 4. Page: 2 Notes by Bibiana Lopez