Question

Factorise 12-(x+x^2) (8-x-x^2)

Answer 1

Answer:

( x - 1 ) ( x - 2 ) ( x + 2 ) ( x + 3 )

Step-by-step explanation:

Expanding first:

  12 - ( x + x² ) ( 8 - x - x² )

= 12 - ( 8x - x² - x³ + 8x² - x³ - x⁴ )

= x⁴ + 2x³ - 7x² - 8x + 12

In the hope of finding some simple factors, try some small integer values for x:

x = 1  =>  x⁴ + 2x³ - 7x² - 8x + 12 = 1 + 2 - 7 - 8 + 12 = 0  =>  ( x - 1 ) is a factor!

Factoring out ( x - 1 ) gives:

   x⁴ + 2x³ - 7x² - 8x + 12 = ( x - 1 ) ( x³ + 3x² - 4x - 12 )

Continuing the search for small factors:

x = 1  =>  x³ + 3x² - 4x - 12 = 1 + 3 - 4 - 12 ≠ 0, so this does not yield a factor.

x = 2 =>  x³ + 3x² - 4x - 12 = 8 + 12 - 8 - 12 = 0  =>  ( x - 2 ) is a factor!

Factoring out ( x - 2 ) gives:

  x⁴ + 2x³ - 7x² - 8x + 12 = ( x - 1 ) ( x - 2 ) ( x² + 5x + 6 )

Now we can use our usual prowess at factoring quadratics to finish this off:

  x⁴ + 2x³ - 7x² - 8x + 12 = ( x - 1 ) ( x - 2 ) ( x + 2 ) ( x + 3 )