How do you find the zeroes of f(x)=x3+10x2−13x−22?

1 Answer
George C.
Aug 5, 2015

Use the rational roots theorem to find possible roots, evaluate f(x) for those candidates to find roots x=−1 and x=2. Deduce the third root x=−11.

Explanation:

By the rational roots theorem, any rational roots of f(x)=0 must be of the form pq where p, q are integers, q≠0, p a divisor of the constant term 22 and q a divisor of the coefficient 1 of the term of highest degree x3.

So the only possible rational roots are:

±1, ±2, ±11, ±22

f(1)=1+10−13−22=−24
f(−1)=−1+10+13−22=0
f(2)=8+40−26−22=0

So far, that gives (x+1) and (x−2) as factors of f(x).

So the remaining factor must be (x+11) to get the correct coefficient for x3 and the constant term −22.

f(x)=(x+1)(x−2)(x+11)

Let us check:

f(−11)=−1331+1210−143−22=0

So the roots of f(x)=0 are x=−1, x=2 and x=−11

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