# How do you find the number of complex, real and rational roots of #x^3-x^2-2x+7=0#?

##### 1 Answer

Use the discriminant to find that it has

#### Explanation:

The cubic equation:

#x^3-x^2-2x+7 = 0#

is in the standard form:

#ax^3+bx^2+cx+d = 0#

Its discriminant

#Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd#

In our example,

#Delta = 4+32+28-1323+252 = -1007#

Since

We were also asked if the roots were rational.

By the rational roots theorem, any rational zeros of this cubic must be expressible in the form

That means that the only possible rational roots are:

#+-1, +-7#

None of these work, so our cubic has no rational roots.