Question

How do you factor $ {x^3} + 3{x^2} - 4x - 12 $

Answer 1

Hint: First we will factorise and evaluate the greatest common factor from the given expression. Then we will factorise the greatest common factor. Factorise the terms until further factorisation cannot be done.

Complete step-by-step answer:
First we will start by factoring out the greatest common factor from each group.
Hence, the equation will become,
 $ {x^2}(x - 3) - 4(x - 3) $
Now we will factorise the polynomial by factoring out the greatest common factor that is
 $ x - 3 $ .
Therefore, the equation becomes, $ (x - 3)({x^2} - 4) $ .
Now we will factorise further.
Here, we can write $ ({x^2} - 4) $ as,
 $
   = ({x^2} - 4) \\
   = ({x^2} - {2^2}) \\
   = (x + 2)(x - 2) \;
  $
Now the equation will become $ (x - 3)(x - 2)(x + 2) $
Now as you can see this expression cannot be factored any further so we will stop factorising here.
Hence, the factors will be $ {x^3} + 3{x^2} - 4x - 12 $ are $ (x - 3)(x - 2)(x + 2) $
So, the correct answer is “ $ (x - 3)(x - 2)(x + 2) $ ”.

Note: Factorising is the reverse of expanding brackets. It is an important way to solve quadratic equations. While factorising an expression you take out any common factors which the terms have. A meaningful factorisation for a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. It may occur to you that all terms of a sum are products and some factors are common to all terms In this case, the distributive law allows factoring out this common factor.
While taking the greatest common factor make sure that the term is present everywhere. You must pay attention to the signs of the terms while taking the terms common. After the final factorisation, you should check if the terms can be factored further or not. Also, check if the factors which you have already evaluated can be factored any further.