How do you simplify an equation from cubic to quadratic?

Question

directrix · Answer

I am unclear on what the question means.

A cubic equation can be supressed into a quadric.
Consider y = x^3 -1. It can be factored to the following:
y= (x-1)(x^2 +x + 1), the product of a linear equation and a quadratic.

anonymous · Answer

thats what im looking for, i am unclear as to how to do that.

directrix · Answer

I made up the cubic equation. Sometimes, a cubic may not factor over the set of Reals. Descartes' Rule of Signs and the Rational Root Theorem come to mind. To my thinking, the supression or depression of the cubic, if possible, is determined by the nature of the given cubic equation.

anonymous · Answer

well if i had $$x^{3}+x^{2}-x-1$$ what would I do with it?

directrix · Answer

Do you know the Rational Root Theorem?

directrix · Answer

I think this factors by grouping.

x^3 + x^2 -x -1

=x^2(x + 1) - (x + 1)

= (x+1)(x^2 -1)

= (x+1)(x+1)(x-1)

Check my work.

anonymous · Answer

nope, i am clueless as to how to do any of the simplifying of cubics

anonymous · Answer

it is right btw

directrix · Answer

This is factoring by grouping which is not restricted to cubics. It is the Distributive Property in reverse.

 The formula for factoring the sum and difference of two cubes are ones I memorized.

Look up special factoring patterns and study them.

anonymous · Answer

will do thanks