How do you factor y= x^4 + 5x^2 - 6 ?

2 Answers
Dec 5, 2015

y=(x^2+6)(x+1)(x-1)

Explanation:

Given y=x^4+5x^2-6

We notice that all terms containing the variable x have x^2 as a factor,
so to simplify things initially we will replace x^2 with z

color(white)("XXX")y=z^2+5z-6
which can easily be factored as:
color(white)("XXX")y=(z+6)(z-1)

Restoring x^2 back in place of z
color(white)("XXX")y=(x^2+6)(x^2-1)

The first factor, (x^2+6), has no obvious sub-factors
but the second, (x^2-1), is the difference of squares with sub-factors (x+1)(x-1)

giving
color(white)("XXX")y=(x^2+6)(x+1)(x-1)

Dec 5, 2015

y = (x^2 + 6)(x+1)(x-1)

Explanation:

Perhaps it will help if I rewrite this expression for y in a slightly different way:

y = (x^2)^2 + 5(x^2) - 6

Let u = x^2. Then we have

y = u^2 + 5u - 6

This is something we can factor pretty easily.

y = (u + 6)(u - 1)

Just substitute back to get y in terms of x:

y = (x^2 + 6)(x^2 - 1)

And now, the last step is to factor the second bit which is a difference of two squares:

y = (x^2 + 6)(x - 1)(x+1)