Question

How do you factor $4{x^2} - 4x - 15?$

Answer 1

Hint: To factor the given expression use sum product method in which the middle term or the coefficient of the variable with degree one, is split in to two terms in such a way that their product will be equals to the product of the first and the last term. First term means a term containing a variable with two degrees and the last term means the constant term.

Complete step-by-step solution:
In order to factorize the given expression $4{x^2} - 4x - 15$ we will use the sum product method for which we have to first find the factors of the product of the first and the last term, that is $4\;{\text{and}}\;( - 15)$ respectively.
$4 \times ( - 15) = - 60$
Now factors of $ - 60$ will be given as
$ - 60 = - 1 \times 2 \times 2 \times 3 \times 5$
From above factors of $ - 60$ we can see that if we take $( - 1 \times 2 \times 5) = - 10\;{\text{and}}\;(2 \times 3) = 6$ as split terms then they will also satisfy the condition that sum of split term should be equal to middle term i.e. $ - 10 + 6 = - 4$
Now coming to the expression,
$ = 4{x^2} - 4x - 15$
Splitting the middle term as the sum of $ - 10\;{\text{and}}\;6$, we will get
$ = 4{x^2} - 10x + 6x - 15$
Now taking out common terms from the first two and the second two terms,
$
   = 2x(2x - 5) + 3(2x - 5) \\
   = (2x - 5)(2x + 3) \\
 $
Therefore $(2x - 5)\;{\text{and}}\;(2x + 3)$ are the required factors of the expression $4{x^2} - 4x - 15$

Note: Sum product method can be only used to factor two degree polynomials or say quadratic expressions. When splitting the middle term, take care of the signs. This problem can be solved by one more method called a long division method, in which you have to find the first factor by hit and trial method, and then just divide the expression with that factor. Long division method is useful in factoring all types of polynomials.