# How do you find the zeros of #y=x^3-2x^2-11x+12#?

##### 1 Answer

The zeros are

#### Explanation:

Given:

#x^3-2x^2-11x+12#

**Sum of coefficients shortcut**

Note that the sum of the coefficients is

#1-2-11+12 = 0#

Hence,

#x^3-2x^2-11x+12 = (x-1)(x^2-x-12)#

**Fishing for factors**

To factor the remaining quadratic we want a pair of factors of

#x^2-x-12 = (x-4)(x+3)#

So the zeros of the given cubic are

So the given example cubic proved to be not too difficult, but what might we do if the sum of coefficients shortcut didn't apply?

**Rational zeros theorem**

Given any polynomial with integer coefficients in standard form, any rational zeros are expressible in the form

In our example, any rational zeros would take the form

#+-1, +-2, +-3, +-4, +-6, +-12#

We could try substituting each in turn until we find a zero, then divide the polynomial by the corresponding factor as before.