Question

Simplify the expression by factoring, showing the steps in your work.

Answer 1

Given:

$\frac{12 x^{2}+31 x+7}{16 x^{2}+8 x+1}$

To find:

The simplified expression by factoring.

Solution:

Let us factor the numerator:

$12 x^{2}+31 x+7$

31x can be written as 3x + 28x.

    $=\left(12 x^{2}+3 x\right)+(28 x+7)$

Take 3x common in 1st two terms and 7 common in next two terms.

    $=3 x(4 x+1)+7(4 x+1)$

Make sure the remaining terms in the both brackets must be same.

Now, take out common factor (4x + 1).

    $=(4 x+1)(3 x+7)$

In the same way, factorize the denominator:

$16 x^{2}+8 x+1$

8x can be written as 4x + 4x.

     $=(16 x^{2}+4 x)+(4x+1)$

Take 4x common in 1st two terms and 1 common in next two terms.

     $=4x(4 x+1)+1(4x+1)$

Make sure the remaining terms in the both brackets must be same.

Now, take out common factor (4x + 1).

     $=(4x+1)(4x+1)$

Substitute the terms we found for numerator and denominator:

$\frac{12 x^{2}+31 x+7}{16 x^{2}+8 x+1}=\frac{(4 x+1)(3 x+7)}{(4 x+1)(4 x+1)}$

Cancel the common factors.

                        $=\frac{3 x+7}{4 x+1}$

The simplified expression is $\frac{3 x+7}{4 x+1}$.