Question 893421: List the potential rational zeros of the polynomial function. Do not find the zeros.
f(x) = x^5 - 4x^2 + 2x + 14
a. ± 1, ±1/7 , ±1/2 , ± 1/14
b. ± 1, ± 7, ± 2, ± 14
c. ± 1, ± 7, ± 2
d. ± 1, ±1/7 , ±1/2 , ± 1/14, ± 7, ± 2, ± 14
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your potential rational zeroes will be any of the factors of the constant term divided by the factors of the leading coefficient.
your equation is x^5 - 4x^2 + 2x + 14
the coefficient of the leading term is 1.
the constant term is 14.
the factors of 14 are +/- 1,2,7,14 because 1*14 = 14 and 2*7 = 14.
the factors of 1 are +/- 1 because 1*1 = 1.
your possible zeroes of this equation are +/- {1/1, 2/1, 7/1, 14/1}.
simplify these to get possible zeroes of +/- {1, 2, 7, 14}
that would be selection b.
in fact, your equation doesn't have any rational zeroes.
it has one zero at 1.334 which is not rational.
if you used synthetic division to solve for a root of your equation, you would find that none of the potential rational zeroes worked.
a graph of your equation looks like this:
an equation where the rational roots will be found is x^2 + 8x - 20
the roots will be found at x = -10 and x = 2 because the factors of this equation are (x + 10) * (x - 2) = 0.
the potential rational roots are +/- {20/1, 10/1, 5/1, 2/1}
the actual roots are at x = -10 and x = 2
the following graph of this equation shows the roots.
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