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2.4 – Zeros of Polynomial Functions • Rational Zero Theorem: If f is a polynomial function of the form f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0 with degree n > 1, integer coefficients, and a0 ≠ 0, then every rational zero of f has the form p/q, where p is all possible factors of a0 and q is all possible factors of an. Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros. g(x) = x4 + 4x3 – 12x – 9 Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros. g(x) = x4 + 4x3 – 12x – 9 p = ±9, ±3, ±1 = ±9, ±3, ±1 (six possible) q ±1 Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros. g(x) = x4 + 4x3 – 12x – 9 p = ±9, ±3, ±1 = ±9, ±3, ±1 (six possible) q ±1 Test all possibilities using the Factor Theorem. Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros. g(x) = x4 + 4x3 – 12x – 9 p = ±9, ±3, ±1 = ±9, ±3, ±1 (six possible) q ±1 Test all possibilities using the Factor Theorem. g(9) = (9)4 + 4(9)3 – 12(9) – 9 g(-9) = (-9)4 + 4(-9)3 – 12(-9) – 9 g(3) = (3)4 + 4(3)3 – 12(3) – 9 g(-3) = (-3)4 + 4(-3)3 – 12(-3) – 9 g(1) = (1)4 + 4(1)3 – 12(1) – 9 g(-1) = (-1)4 + 4(-1)3 – 12(-1) – 9 Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros. g(x) = x4 + 4x3 – 12x – 9 p = ±9, ±3, ±1 = ±9, ±3, ±1 (six possible) q ±1 Test all possibilities using the Factor Theorem. g(9) = (9)4 + 4(9)3 – 12(9) – 9 = 9360 g(-9) = (-9)4 + 4(-9)3 – 12(-9) – 9 = 3744 g(3) = (3)4 + 4(3)3 – 12(3) – 9 = 144 g(-3) = (-3)4 + 4(-3)3 – 12(-3) – 9 = 0 g(1) = (1)4 + 4(1)3 – 12(1) – 9 = -16 g(-1) = (-1)4 + 4(-1)3 – 12(-1) – 9 = 0 Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0 Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0 *Find real zeros using graphing calculator. Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0 *Find real zeros using graphing calculator. x = -2, x = -1, x = 3 Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0 *Find real zeros using graphing calculator. x = -2, x = -1, x = 3 *Divide to find remaining zero. Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0 *Find real zeros using graphing calculator. x = -2, x = -1, x = 3 *Divide to find remaining zero. -2| 1 2 -7 -20 -12 | Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0 *Find real zeros using graphing calculator. x = -2, x = -1, x = 3 *Divide to find remaining zero. -2| 1 2 -7 -20 -12 -2 0 14 12 1 0 -7 -6 | 0 x3 – 7x – 6 -1| 1 0 -7 -6 | -1| 1 1 0 -7 -6 -1 1 6 -1 -6 | 0 x2 – x – 6 -1| 1 1 0 -7 -6 -1 1 6 -1 -6 | 0 x2 – x – 6 So (x + 2)(x + 1)(x2 – x – 6) = 0 -1| 1 1 0 -7 -6 -1 1 6 -1 -6 | 0 x2 – x – 6 So (x + 2)(x + 1)(x2 – x – 6) = 0 (x + 2)(x + 1)(x – 3)(x + 2) = 0 -1| 1 1 0 -7 -6 -1 1 6 -1 -6 | 0 x2 – x – 6 So (x + 2)(x + 1)(x2 – x – 6) = 0 (x + 2)(x + 1)(x – 3)(x + 2) = 0 (x + 1)(x – 3)(x + 2)2 = 0 -1| 1 1 0 -7 -6 -1 1 6 -1 -6 | 0 x2 – x – 6 So (x + 2)(x + 1)(x2 – x – 6) = 0 (x + 2)(x + 1)(x – 3)(x + 2) = 0 (x + 1)(x – 3)(x + 2)2 = 0 So x = -1, x = 3, and x = -2 (twice). Ex. 3 Write each function as (a) the product of linear and irreducible factors and (b) the product of linear factors. Then (c) list all of its zeros. g(x) = x4 – 3x3 – 12x2 + 20x + 48 Ex. 3 Write each function as (a) the product of linear and irreducible factors and (b) the product of linear factors. Then (c) list all of its zeros. g(x) = x4 – 3x3 – 12x2 + 20x + 48 *Find real zeros on graphing calculator. Ex. 3 Write each function as (a) the product of linear and irreducible factors and (b) the product of linear factors. Then (c) list all of its zeros. g(x) = x4 – 3x3 – 12x2 + 20x + 48 *Find real zeros on graphing calculator. x = -2, x = 3, x = 4 Ex. 3 Write each function as (a) the product of linear and irreducible factors and (b) the product of linear factors. Then (c) list all of its zeros. g(x) = x4 – 3x3 – 12x2 + 20x + 48 *Find real zeros on graphing calculator. x = -2, x = 3, x = 4 *Divide to find remaining zero. Ex. 3 Write each function as (a) the product of linear and irreducible factors and (b) the product of linear factors. Then (c) list all of its zeros. g(x) = x4 – 3x3 – 12x2 + 20x + 48 *Find real zeros on graphing calculator. x = -2, x = 3, x = 4 *Divide to find remaining zero. -2| 1 -3 -12 20 48 | Ex. 3 Write each function as (a) the product of linear and irreducible factors and (b) the product of linear factors. Then (c) list all of its zeros. g(x) = x4 – 3x3 – 12x2 + 20x + 48 *Find real zeros on graphing calculator. x = -2, x = 3, x = 4 *Divide to find remaining zero. -2| 1 -3 -12 20 48 -2 10 4 -48 1 -5 -2 24 | 0 3| 1 -5 -2 24 | 3| 1 1 -5 3 -2 -2 -6 -8 24 -24 |0 3| 1 -5 -2 3 -6 1 -2 -8 x2 – 2x – 8 24 -24 |0 3| 1 -5 -2 3 -6 1 -2 -8 x2 – 2x – 8 24 -24 |0 So g(x) = (x + 2)(x – 3)(x2 – 2x – 8) 3| 1 -5 -2 3 -6 1 -2 -8 x2 – 2x – 8 24 -24 |0 So g(x) = (x + 2)(x – 3)(x2 – 2x – 8) = (x + 2)(x – 3)(x + 2)(x – 4) 3| 1 -5 -2 3 -6 1 -2 -8 x2 – 2x – 8 24 -24 |0 So g(x) = (x + 2)(x – 3)(x2 – 2x – 8) = (x + 2)(x – 3)(x + 2)(x – 4) = (x + 2)2(x – 3)(x – 4) 3| 1 -5 -2 3 -6 1 -2 -8 x2 – 2x – 8 24 -24 |0 So g(x) = (x + 2)(x – 3)(x2 – 2x – 8) = (x + 2)(x – 3)(x + 2)(x – 4) (a) & (b) = (x + 2)2(x – 3)(x – 4) (c) x = -2 (twice), x = 3, x = 4 Ex. 4 Use the given zero to find all complex zeros of each function. Then write the linear factorization of the function. h(x) = 3x5 – 5x4 – 13x3 – 65x2 – 2200x + 1500; -5i Ex. 4 Use the given zero to find all complex zeros of each function. Then write the linear factorization of the function. h(x) = 3x5 – 5x4 – 13x3 – 65x2 – 2200x + 1500; -5i -5i| 3 -5 -13 -65 -2200 1500 | Ex. 4 Use the given zero to find all complex zeros of each function. Then write the linear factorization of the function. h(x) = 3x5 – 5x4 – 13x3 – 65x2 – 2200x + 1500; -5i -5i| 3 -5 -15i 3 -5-15i -13 -65 -2200 -75+25i 125+440i 2200-300i -88+25i 60+440i -300i 1500 -1500 |0 Ex. 4 Use the given zero to find all complex zeros of each function. Then write the linear factorization of the function. h(x) = 3x5 – 5x4 – 13x3 – 65x2 – 2200x + 1500; -5i -5i| 3 -5 -15i 3 -5-15i 5i| 3 -5-15i -13 -65 -2200 -75+25i 125+440i 2200-300i -88+25i 60+440i -300i -88+25i 60+440i -300i | 1500 -1500 |0 Ex. 4 Use the given zero to find all complex zeros of each function. Then write the linear factorization of the function. h(x) = 3x5 – 5x4 – 13x3 – 65x2 – 2200x + 1500; -5i -5i| 3 -5 -15i 3 -5-15i -13 -65 -2200 -75+25i 125+440i 2200-300i -88+25i 60+440i -300i 5i| 3 -5-15i -88+25i 60+440i +15i -25i -440i 3 -5 -88 60 -300i 300i | 0 1500 -1500 |0 3 -5 -88 60 3x3 – 5x2 – 88x + 60 3 -5 -88 60 3x3 – 5x2 – 88x + 60 *Find real zeros on graphing calculator. 3 -5 -88 60 3x3 – 5x2 – 88x + 60 *Find real zeros on graphing calculator. x = 5, x = 6, x = 2/3 3 -5 -88 60 3x3 – 5x2 – 88x + 60 *Find real zeros on graphing calculator. x = 5, x = 6, x = 2/3 So x = -5i, x = 5i, x = 5, x = 6, x = 2/3 3 -5 -88 60 3x3 – 5x2 – 88x + 60 *Find real zeros on graphing calculator. x = 5, x = 6, x = 2/3 So x = -5i, x = 5i, x = 5, x = 6, x = 2/3 And (x + 5i)(x – 5i)(x – 5)(x – 6)(x – 2/3) 3 -5 -88 60 3x3 – 5x2 – 88x + 60 *Find real zeros on graphing calculator. x = 5, x = 6, x = 2/3 So x = -5i, x = 5i, x = 5, x = 6, x = 2/3 And (x + 5i)(x – 5i)(x – 5)(x – 6)(x – 2/3) or (x + 5i)(x – 5i)(x – 5)(x – 6)(3x – 2)