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sleet_krkn [62]
3 years ago
12

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

Mathematics
1 answer:
alexdok [17]3 years ago
3 0

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}.

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