Question

How do you use the Rational Root Theorem to find all the roots of `x^3 + 9x^2 + 19x – 4 = 0`?

Answer 1

The roots are: `-4` and `(-5+-sqrt(29))/2`

Explanation:

`f(x) = x^3+9x^2+19x-4`

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 factor of the constant term `-4` and `q` a divisor of the coefficient `1` of the leading term.

That means that the only possible rational zeros are:

`+-1, +-2, +-4`

Trying each in turn, we find:

`f(-4) = -64+9(16)+19(-4)-4 = -64+144-76-4 = 0`

So `x=-4` is a zero and `(x+4)` a factor:

`x^3+9x^2+19x-4 = (x+4)(x^2+5x-1)`

We can solve the remaining quadratic using the quadratic formula:

`x = (-5+-sqrt(5^2-4(1)(-1)))/(2*1)`

`=(-5+-sqrt(29))/2`