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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)

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