How do you use the rational roots theorem to find all possible zeros of f(x)=3x3−17x2+18x+8?

1 Answer
George C.
Jun 5, 2016

x=−13, x=2, x=4

Explanation:

f(x)=3x3−17x2+18x+8

By the rational root theorem, any rational zeros of f(x) must be expressible in the form pq for integers p,q with p a divisor of the constant term 8 and q a divisor of the coefficient 3 of the leading term.

That means that the only possible rational zeros are:

±13, ±23, ±1, ±43, ±2, ±83, ±4, ±8

We find:

f(−13)=−19−179−6+8=0

So x=−13 is a zero and (3x+1) a factor of f(x)

3x3−17x2+18x+8

=(3x+1)(x2−6x+8)

=(3x+1)(x−2)(x−4)

Hence the other two zeros are x=2 and x=4

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