Question

How do you solve `x^3+3x^2-10x-150` using the fundamental theorem of algebra?

Answer 1

Three roots of `x^3+3x^2-10x-150` are `{5,-4-isqrt14,-4+isqrt14)`

Explanation:

According to fundamental theorem of algebra, any polynomial of degree `n` with complex coefficients has exactly `n` roots, counted with multiplicity.

As `f(x)=x^3+3x^2-10x-150` is a polynomial of degree `3`, it has `n` roots.

It is apparent that `5` is one of the root as

`f(5)=5^3+3xx5^2-10xx5-150=125+75-50-150=0` and hence `(x-5)` divides `x^3+3x^2-10x-150`.

Dividing by `(x-5)`, we get `x^2+8x+30` or

`x^3+3x^2-10x-150=(x-5)(x^2+8x+30)`

using quadratic formula complex roots of `x^2+8x+30` are `(-8+-sqrt(8^2-4xx1xx30))/(2xx1)` or `(-8+-sqrt(64-120))/2=-4+-isqrt56/2`

or `-4+-isqrt14`

Hence, three roots of `f(x)=x^3+3x^2-10x-150` are `{5,-4-isqrt14,-4+isqrt14)`