5.6 Find Rational Zeros

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5.6 Find Rational Zeros
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The polynomial function
Has zeros
Notice the numerators of these zeros (-3, -5, and
7) are factors of the constant term 105.
Notice the denominators of these zeros (2, 4, 8)
are factors of the leading coefficient 64.
The Rational Zero Theorem
If
Has integer coefficients, then every rational
zero of has has the following form
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Since the coefficients are all integers, we can
us the rational zero theorem.
Find the rational zeros of
Factors of 12 1, 2, 3, 4, 6, 12
Factors of 1 1
Test x 1
Test x -1
1 2 -11 -12
1 2 -11 -12
1
-1
1 3 -8
-1 -1 12
1 3 -8 -20
1 1 -12 0
Since 1 is a zero of f, you can write the
following
The zeros of f(x) are 1, 3, and 4.
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Given
The lengthy list of possible rational zeros
With so many possibilities, it is worth our time
to sketch the graph of the function.
From the graph we can eliminate some choices and
test others that seem reasonable. Reasonable
choices are
Use these two given zeros and continue finding
the other zeros.