Question

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

Answer 1

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.

and

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.

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`.