Question

How do you factor the expression ${x^2} + 14x + 48?$

Answer 1

Hint:Use sum product method to factor the given expression, split the coefficient of $x$ in such a manner that their product should be equals to the product of the coefficient of ${x^2}$ and the constant in the expression. Then take the common terms out and factorize the expression.

Complete step by step solution:
In order to factorize the expression ${x^2} + 14x + 48$ we should go through the sum product method, in which we have to split the middle term $\left( {{\text{coefficient}}\;{\text{of}}\;x\;(14)} \right)$ in such a manner that their sum and product should respectively equals to the coefficient of $x$ and the product of the coefficient of ${x^2}\;(1)$ and the constant $(48)$.

Since the product of the coefficient of ${x^2}\;(1)$ and the constant $(48)\; = 1 \times 48 = 48$

Therefore we have to find the factors of $48$ in order to split the middle term $\left(
{{\text{coefficient}}\;{\text{of}}\;x\;(14)} \right)$

Factors of $48$ can be written as
$48 = 2 \times 2 \times 2 \times 2 \times 3$

See the factors, if we take as $2 \times 2 \times 2 = 8\;{\text{and}}\;2 \times 3 = 6$ then this will satisfy the desired conditions, that is their sum $(8 + 6 = 14)$ equals the middle term $\left( {{\text{coefficient}}\;{\text{of}}\;x\;(14)} \right)$ and product $(8 \times 6 = 48)$ equals the product of the coefficient of ${x^2}\;(1)$ and the constant $(48)$

So splitting the middle term in the expression, we will get
$
= {x^2} + 14x + 48 \\
= {x^2} + 8x + 6x + 48 \\
$
Taking $x$ common from ${x^2} + 8x$ and $6$ common from $6x + 48$
$
= {x^2} + 8x + 6x + 48 \\
= x(x + 8) + 6(x + 8) \\
$
Again taking $(x + 8)$ common from $x(x + 8) + 6(x + 8)$
$
= x(x + 8) + 6(x + 8) \\
= (x + 8)(x \times 1 + 6 \times 1) \\
= (x + 8)(x + 6) \\
$
Therefore the required factors of \[{x^2} + 14x + 48\;{\text{are}}\;(x + 8)\;{\text{and}}\;(x + 6)\]

Note: Sometimes it is difficult to factorize using the sum product method when the digits are bigger or the degree of the expression is greater. In that type of questions try to find common factors among all the terms of the expression and after taking the common factor out, apply the sum product method.