Question

How do you find the roots of `x^3+x^2-7x+2=0`?

Answer 1

The roots are `2` and `-3/2 +- sqrt(13)/2`

Explanation:

Given:

`f(x) = x^3+x^2-7x+2`

By the rational roots theorem, any rational zeros of `f(x)` are expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `2` and `q` a divisor of the coefficient `1` of the leading term.

That means that the only possible rational zeros are:

`+-1, +-2`

Note that:

`f(2) = 8+4-14+2 = 0`

So `x=2` is a zero and `(x-2)` a factor:

`x^3+x^2-7x+2 = (x-2)(x^2+3x-1)`

We can find the zeros of the remaining quadratic factor using the quadratic formula with `a=1`, `b=3` and `c=-1` :

`x = (-color(blue)(3) +-sqrt (color(blue)(3)^2 - 4(color(blue)(1))(color(blue)(-1))))/(2(color(blue)(1)))`

`color(white)(x) = (-3+-sqrt(9+4))/2`

`color(white)(x) = -3/2+-sqrt(13)/2`