Question

Factorise:

x⁶ + 26x³ – 27

Answer 1

To factorise x⁶ + 26x³ – 27, notice that we can treat this as a quadratic expression in terms of x³.

Let y = x³.

Then the expression becomes:

y² + 26y – 27

Now, factor the quadratic expression:

y² + 26y – 27 = (y + 27) (y – 1)

Substitute y = x³ back into the factored expression:

(x³ + 27) (x³ – 1)

Next, factor each of these terms further.

Both are differences and sums of cubes:

x³ + 27 = (x + 3) (x² – 3x + 9)

x³ – 1 = (x – 1) (x² + x + 1)

Thus, the fully factorized form of x⁶ + 26x³ – 27 is (x + 3) (x – l) (x² – 3x + 9) (x² + x + 1)