6.5 The Remainder and Factor Theorems Goals p Divide polynomials and relate the result to the remainder theorem and the factor theorem. p Use polynomial division in real-life problems. Your Notes VOCABULARY Polynomial long division A method used to divide polynomials similar to the way you divide numbers Synthetic division A method used to divide a polynomial by an expression of the form x Ϫ k Example 1 Using Polynomial Long Division .Divide 4x4 ؊ x2 ؊ 18x ؉ 8 by x2 ؉ 2x ؉ 3 Write division in the same format you would use when dividing numbers. Include a “0” as the coefficient of x3. 4 Ϫ 3 2 4x 8x 3x x2 x2 x2 4x2 Ϫ 8x ϩ 3 x2 ϩ 2x ϩ 3ͤ4ෆxෆ4ෆϩෆ0xෆ3ෆϪෆ1ෆ2ෆxෆ2ෆϪෆෆ18ෆxෆϩෆෆ8 4x4 ϩ 8x3 ϩ 12x2 Ϫ8x3 Ϫ 13x2 Ϫ 18x Ϫ8x3 Ϫ 16x2 Ϫ 24x 3x2 ϩ 6x ϩ 8 3x2 ϩ 6x ϩ 9 Ϫ1 Write the result as follows. 4x4 Ϫ x2 Ϫ 18x ϩ 8 Ϫ1 ϭ 4x2 Ϫ 8x ϩ 3 ϩ x2 ϩ 2x ϩ 3 x2 ϩ 2x ϩ 3 136 Algebra 2 Notetaking Guide • Chapter 6 Your Notes REMAINDER THEOREM If a polynomial f(x) is divided by x Ϫ k, the remainder is r ϭ f(k) . Example 2 Using Synthetic Division Divide x3 ϩ x2 Ϫ 5x ϩ 3 by x ϩ 2. Solution To find the value of k, rewrite the divisor in the form x Ϫ k. Because x ϩ 2 ϭ x Ϫ (Ϫ2) , k Ϫ2 . Ϫ2 11Ϫ53 Ϫ226 1 Ϫ1 Ϫ39 x3 ϩ x2 Ϫ 5x ϩ 3 9 ᎏᎏᎏ x2 Ϫ x Ϫ 3 ϩ x ϩ 2 x 2 FACTOR THEOREM A polynomial f(x) has a factor x Ϫ k if and only if f(k) ϭ 0 . Example 3 Factoring a Polynomial Factor f(x) x3 Ϫ 19x Ϫ 30 given that f(5) 0. Solution Because f(5) ϭ 0, you know that x Ϫ 5 is a factor of f(x). Use synthetic division to find the other factors. 5 10Ϫ19 Ϫ30 52530 15 6 0 The result gives the coefficients of the quotient. x3 Ϫ 19x Ϫ 30 ϭ ( x Ϫ 5 )( x2 ϩ 5x ϩ 6 ) ϭ ( x Ϫ 5 )( x ϩ 2 )( x ϩ 3 ) Lesson 6.5 • Algebra 2 Notetaking Guide 137 Your Notes Example 4 Finding Zeros of a Polynomial Function A zero of f(x) ϭ x3 ϩ x2 Ϫ 4x Ϫ 4 is x ϭϪ1. Find the other zeros. Solution Because f(Ϫ1) ϭ 0, you know that x ϩ 1 is a factor of f(x). Use synthetic division to find the other factors. Ϫ1 11Ϫ4 Ϫ4 Ϫ104 10Ϫ40 The result gives the coefficients of the quotient. f(x) ϭ x3 ϩ x2 Ϫ 4x Ϫ 4 ϭ ( x ϩ 1 )( x2 Ϫ 4 ) ϭ ( x ϩ 1 )( x ϩ 2 )( x Ϫ 2 ) By the factor theorem, the zeros of f are Ϫ1, Ϫ2, and 2 . Checkpoint Complete the following exercises. 1. Use long division to divide x2 ϩ 4x Ϫ 1 by x ϩ 3. Ϫ4 x ϩ 1 ϩ x ϩ 3 2. Use synthetic division to divide 2x3 Ϫ x2 ϩ 3x ϩ 4 by x ϩ 1. Ϫ2 2x2 Ϫ 3x 6 ϩ x ϩ 1 3. Factor f(x) ϭ 2x3 Ϫ x2 Ϫ 25x Ϫ 12 given that f(4) ϭ 0. Homework (x Ϫ 4)(x ϩ 3)(2x ϩ 1) 4. A zero of f(x) ϭ x4 5x2 ϩ 4 is 1. Find the other zeros. Ϫ1, Ϫ2, 2 138 Algebra 2 Notetaking Guide • Chapter 6.