How do you factor y=x4+5x26 ?

2 Answers
Dec 5, 2015

y=(x2+6)(x+1)(x1)

Explanation:

Given y=x4+5x26

We notice that all terms containing the variable x have x2 as a factor,
so to simplify things initially we will replace x2 with z

XXXy=z2+5z6
which can easily be factored as:
XXXy=(z+6)(z1)

Restoring x2 back in place of z
XXXy=(x2+6)(x21)

The first factor, (x2+6), has no obvious sub-factors
but the second, (x21), is the difference of squares with sub-factors (x+1)(x1)

giving
XXXy=(x2+6)(x+1)(x1)

Dec 5, 2015

y=(x2+6)(x+1)(x1)

Explanation:

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

y=(x2)2+5(x2)6

Let u=x2. Then we have

y=u2+5u6

This is something we can factor pretty easily.

y=(u+6)(u1)

Just substitute back to get y in terms of x:

y=(x2+6)(x21)

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

y=(x2+6)(x1)(x+1)