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

1 Answer
Jul 24, 2015

Answer:

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

Explanation:

According to the rational root theorem, the rational roots of #f(x) = 0# must all be of the form #p/q# 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) = x^2 -3x-10# is also a factor.

We can use synthetic division to find the other factor.

3

The other factor is #x^2 + 1#.

#x^2+1=0#
#x^2=-1#
#x=±sqrt(-1) = ±i#

#x=-i# or #x=i#

So

#f(x) = x^4-3x^3-9x^2-3x-10 = (x+2)(x-5)(x^2+1)#.

and

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