How do you use synthetic division to find the zeroes of f(x)= x^4 -3x^3-9x^2-3x-10f(x)=x4−3x3−9x2−3x−10?

1 Answer
Ernest Z.
Jul 24, 2015

The zeroes of f(x) = x^4-3x^3-9x^2-3x-10f(x)=x4−3x3−9x2−3x−10 are color(red)(-2, 5, -i, i)−2,5,−i,i.

Explanation:

According to the rational root theorem, the rational roots of f(x) = 0f(x)=0 must all be of the form p/qpq with pp a divisor of -10−10 and qq a divisor of 11.

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

We have to test all eight possibilities.

Here are the only two that work.

1

and

2

So -2−2 and 55 are zeroes of the polynomial.

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

(x+2)(x-5) = x^2 -3x-10(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 x^2 + 1x2+1.

x^2+1=0x2+1=0
x^2=-1x2=−1
x=±sqrt(-1) = ±ix=±√−1=±i

x=-ix=−i or x=ix=i

So

f(x) = x^4-3x^3-9x^2-3x-10 = (x+2)(x-5)(x^2+1)f(x)=x4−3x3−9x2−3x−10=(x+2)(x−5)(x2+1).

and

The roots of f(x) = x^4-3x^3-9x^2-3x-10f(x)=x4−3x3−9x2−3x−10 are -2, 5, -i, i−2,5,−i,i.

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