Transcript Document
2.4 – Zeros of Polynomial Functions
• Rational Zero Theorem: If f is a polynomial function of the form f(x) = a
n x n + a n-1 x n-1 + … + a 2 x 2 + a 1 x + a 0
with degree n > 1, integer coefficients, and
a 0
≠ 0, then every rational zero of f has the form p/q, where p is all possible factors of
a 0
and q is all possible factors of a
n
.
Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros.
g(x) = x 4 + 4x 3 – 12x – 9
Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros.
g(x) = x 4 + 4x 3 – 12x – 9
p
= ±9, ±3, ±1
q
±1 = ±9, ±3, ±1 (six possible)
Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros.
g(x) = x 4 + 4x 3 – 12x – 9
p q
= ±9, ±3, ±1 ±1 = ±9, ±3, ±1 (six possible) 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) = x 4 + 4x 3 – 12x – 9
p q
= ±9, ±3, ±1 ±1 = ±9, ±3, ±1 (six possible) 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) = x 4 + 4x 3 – 12x – 9
p q
= ±9, ±3, ±1 ±1 = ±9, ±3, ±1 (six possible) 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.
x
4 + 2x 3 – 7x 2 – 20x – 12 = 0
Ex. 2 Solve the equation.
x
4 + 2x 3 – 7x 2 – 20x – 12 = 0 *Find real zeros using graphing calculator.
Ex. 2 Solve the equation.
x
4 + 2x 3 – 7x 2 – 20x – 12 = 0 *Find real zeros using graphing calculator.
x = -2, x = -1, x = 3
Ex. 2 Solve the equation.
x
4 + 2x 3 – 7x 2 – 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.
x
4 + 2x 3 – 7x 2 – 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.
x
4 + 2x 3 – 7x 2 – 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 1 -2 0 0 -7
x
3 – 7x – 6 14 -6 12 | 0
-1| 1 0 -7 -6 |
-1| 1 1 0 -7 -1 1 -1
x
2 -6 – x – 6 -6 6 | 0
-1| 1 1 0 -7 -1 1 -1
x
2 -6 – x – 6 -6 6 | 0 So (x + 2)(x + 1)(x 2 – x – 6) = 0
-1| 1 1 0 -7 -1 1 -1
x
2 -6 – x – 6 -6 6 | 0 So (x + 2)(x + 1)(x 2 – x – 6) = 0 (x + 2)(x + 1)(x – 3)(x + 2) = 0
-1| 1 1 0 -7 -1 1 -1
x
2 -6 – x – 6 -6 6 | 0 So (x + 2)(x + 1)(x 2 – 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 -1 1 -1
x
2 -6 – x – 6 -6 6 | 0 So (x + 2)(x + 1)(x 2 – 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) = x 4 – 3x 3 – 12x 2 + 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) = x 4 – 3x 3 – 12x 2 + 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) = x 4 – 3x 3 – 12x 2 + 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) = x 4 – 3x 3 – 12x 2 + 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) = x 4 – 3x 3 – 12x 2 + 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) = x 4 – 3x 3 – 12x 2 + 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
x
2 – 2x – 8 -8 24 -24 | 0
3| 1 -5 -2 3 -6 1 -2
x
2 – 2x – 8 -8 24 -24 | 0 So g(x) = (x + 2)(x – 3)(x 2 – 2x – 8)
3| 1 -5 -2 3 -6 1 -2
x
2 – 2x – 8 -8 24 -24 | 0 So g(x) = (x + 2)(x – 3)(x 2 – 2x – 8) = (x + 2)(x – 3)(x + 2)(x – 4)
3| 1 -5 -2 3 -6 1 -2
x
2 – 2x – 8 -8 24 -24 | 0 So g(x) = (x + 2)(x – 3)(x 2 – 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
x
2 – 2x – 8 -8 24 -24 | 0 (a) & (b) (c) So g(x) = (x + 2)(x – 3)(x 2 – 2x – 8) = (x + 2)(x – 3)(x + 2)(x – 4) = (x + 2) 2 (x – 3)(x – 4) 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) = 3x 5 – 5x 4 – 13x 3 – 65x 2 – 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) = 3x 5 – 5x 4 – 13x 3 – 65x 2 – 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) = 3x 5 – 5x 4 – 13x 3 – 65x 2 – 2200x + 1500; -5i -5i| 3 -5 -15i 3 -5-15i -13 -65 -2200 1500 -75+25i 125+440i 2200-300i -1500 -88+25i 60+440i -300i | 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) = 3x 5 – 5x 4 – 13x 3 – 65x 2 – 2200x + 1500; -5i -5i| 3 -5 -15i 3 -5-15i -13 -65 -2200 1500 -75+25i 125+440i 2200-300i -1500 -88+25i 60+440i -300i | 0 5i| 3 -5-15i -88+25i 60+440i -300i |
Ex. 4 Use the given zero to find all complex zeros of each function. Then write the linear factorization of the function.
h(x) = 3x 5 – 5x 4 – 13x 3 – 65x 2 – 2200x + 1500; -5i -5i| 3 -5 -15i 3 -5-15i -13 -65 -2200 1500 -75+25i 125+440i 2200-300i -1500 -88+25i 60+440i -300i | 0 5i| 3 3 -5-15i +15i -5 -88+25i 60+440i -25i -440i -88 60 -300i 300i | 0
3 -5 -88 60 3x 3 – 5x 2 – 88x + 60
3 -5 -88 60 3x 3 – 5x 2 – 88x + 60 *Find real zeros on graphing calculator.
3 -5 -88 60 3x 3 – 5x 2 – 88x + 60 *Find real zeros on graphing calculator.
x = 5, x = 6, x = 2/3
3 -5 -88 60 3x 3 – 5x 2 – 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 3x 3 – 5x 2 – 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 3x 3 – 5x 2 – 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)