3.6 The Real Zeros of a Polynomial Function
Objective: Find the real zeros of f and use them to factor f.
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You can follow a step­by­step process to find the real zeros and factor f.
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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
q
+ 1, + 1/2, + 2, + 3, + 3/2, + 6
:
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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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Example: f(x) = 2x3 + 11x2 ­ 7x ­ 6
Step 4: Use long 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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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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Homework: page 231 (40 ­ 44, 47 ­ 48)
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