3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
3.6 The Real Zeros of a Polynomial Function
Objectives: Use the Factor Theorem.
Use the Rational Zeros Theorem.
Find the real zeros of f and use them to factor f.
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
Warm­up: Use synthetic division to divide.
4x6 ­ 64x4 + x2 ­ 16 ÷ x ­ 4
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
Factor Theorem:
Let f be a polynomial function.
Then x ­ c is a factor of f(x) if and only if f(c) = 0.
1. If f(c) = 0, then x ­ c is a factor of f(x).
2. If x ­ c is a factor of f(x), then f(c) = 0.
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
Use the factor theorem to determine whether x ­ c is a factor of f. If it is, write f in factored form.
Ex) f(x) = 3x4 ­ 6x3 ­ 5x + 10; c = 1
4
3
f(1) = 3(1) ­ 6(1) ­ 5(1) + 10
= 3(1) ­ 6(1) ­ 5(1) + 10
= 3 ­ 6 ­ 5 + 10
= 2
Therefore, 1 is not a factor.
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
Use the factor theorem to determine whether x ­ c is a factor of f. If it is, write f in factored form.
Ex) f(x) = 4x6 ­ 64x4 + x2 ­ 16; c = 4
4
2
6
f(4) = 4(4) ­ 64(4) + (4) ­ 16
= 4(4096) ­ 64(256) + (16) ­ 16
= 16,384 ­ 16,384 + (16) ­ 16
= 0 Therefore, 4 is a factor of the function.
Now write f(x) in factored form.
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
Use the factor theorem to determine whether x ­ c is a factor of f. If it is, write f in factored form.
Ex) f(x) = 4x6 ­ 64x4 + x2 ­ 16; c = 4
4
4 0 ­64
0
1
0 ­16
f(x) = (x ­ 4)(4x5 + 16x4 + x + 4)
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
Use the function to tell the maximum number of real zeros that each polynomial function may have.
Ex) f(x) = 3x4 ­ 3x3 + x2 + x + 1
There are 4 possible real zeros.
Ex) f(x) = 3x5 ­ x + 2
There are 5 possible real zeros.
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
List the potential rational zeros of each polynomial function. Ex) f(x) = 3x4 ­ 3x3 + x2 + x + 1
"q"
"p"
Find all the factors of p and q. p
Then find all the ratios of q .
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
List the potential rational zeros of each polynomial function. Ex) f(x) = 3x4 ­ 3x3 + x2 + x + 1
"p"
"q"
p: +_ 1
q: +_ 1,
p
:
q
+
_
+
_
3
1, +_
1
3
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
List the potential rational zeros of each polynomial function. Ex) f(x) = 3x5 ­ x + 2
+
_
p: +_ 1, 2
q: +_ 1,
p
:
q
+
_
+
_
3
+
+
_
_
_ 1 +
1, , 2,
3
2
3
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
Find the real zeros of the polynomial function.
Example: f(x) = 2x3 + 11x2 ­ 7x ­ 6
Step 1: Use the degree of the polynomial to find the maximum number of zeros.
There will be at most 3 real zeros.
Step 2: Use the Rational Zeros Theorem to find potential zeros.
(Find all factors of "p" and "q" and all ratios of p to q.)
p: + 1, + 2, + 3, + 6
q: + 1, + 2
p
1
3
: + 1, + /2, + 2, + 3, + /2, + 6
q
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
Example: f(x) = 2x3 + 11x2 ­ 7x ­ 6
Step 3: Look at the graph on a graphing calculator to find possible zeros.
­6
­ 1/ 2 1
It looks like there are zeros at: ­ 6, ­ 1/2 and 1.
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3.6 The Real Zeros of a Polynomial Function 2011
Example:
April 06, 2011
f(x) = 2x3 + 11x2 ­ 7x ­ 6
Step 4: Use synthetic division to test potential rational zeros based on the graph.
(Keep repeating step 4 to find all zeros.)
­6
2 11 ­7 ­6
­12 6 6
2 ­1 ­1 0
­1/2
2 ­1 ­1 ­1 1 2 ­2 0 1
2 ­2 2 2 0
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
Example: f(x) = 2x3 + 11x2 ­ 7x ­ 6
Step 5: Write f in factored form.
f(x) = 2(x + 6)(x + 1/2)(x ­ 1)
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3.6 The Real Zeros of a Polynomial Function 2011
April 06, 2011
Homework: page 231
(12 ­ 32 even, 40 ­ 44, 47 ­ 48)
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