How do you factor $x^6-26x^3-27$ ?

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Answer 1 · 2015-02-05T04:18:42

First let's notice that $x^6=(x^3)^2$

Let $r=x^3$

Using substitution we have

$r^2-26r-27$

We now look for the factors of $-27$ that will combine (add) to make $-26$

We find that $-27$ and $1$ multiply to be $-27$ and add to be $-26$ . This gives us

$(r-27)(r+1)$

Now substitute $x^3$ back in for $r$

$(x^3-27)(x^3+1)$

We should recognize this as a difference of cubes and a sum of cubes.

$(x-3)(x^2+3x+9)(x+1)(x^2-x+1)$

How do you factor x^6-26x^3-27x^6-26x^3-27?