How do you use synthetic division to find the zeroes of f(x)=x4−3x3−9x2−3x−10?

1 Answer
Ernest Z.
Jul 24, 2015

The zeroes of f(x)=x4−3x3−9x2−3x−10 are −2,5,−i,i.

Explanation:

According to the rational root theorem, the rational roots of f(x)=0 must all be of the form pq with p a divisor of −10 and q a divisor of 1.

So the only possible rational roots are ±1,±2,±5,±10.

We have to test all eight possibilities.

Here are the only two that work.

1

and

2

So −2 and 5 are zeroes of the polynomial.

That means that x+2 and x−5 are factors, and

(x+2)(x−5)=x2−3x−10 is also a factor.

We can use synthetic division to find the other factor.

3

The other factor is x2+1.

x2+1=0
x2=−1
x=±√−1=±i

x=−i or x=i

So

f(x)=x4−3x3−9x2−3x−10=(x+2)(x−5)(x2+1).

and

The roots of f(x)=x4−3x3−9x2−3x−10 are −2,5,−i,i.

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