Simplify (4x^2-7x-2)/(8x^3+2x^2+4x+1)

$\frac{4 x^{2} - 7 x - 2}{8 x^{3} + 2 x^{2} + 4 x + 1}$

Step 1

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 2 = - 8$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 x$ .

$\frac{4 x^{2} - 7 (x) - 2}{8 x^{3} + 2 x^{2} + 4 x + 1}$

Rewrite $- 7$ as $1$ plus $- 8$

$\frac{4 x^{2} + (1 - 8) x - 2}{8 x^{3} + 2 x^{2} + 4 x + 1}$

Apply the distributive property.

$\frac{4 x^{2} + 1 x - 8 x - 2}{8 x^{3} + 2 x^{2} + 4 x + 1}$

Multiply $x$ by $1$ .

$\frac{4 x^{2} + x - 8 x - 2}{8 x^{3} + 2 x^{2} + 4 x + 1}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(4 x^{2} + x) - 8 x - 2}{8 x^{3} + 2 x^{2} + 4 x + 1}$

Factor out the greatest common factor (GCF) from each group.

$\frac{x (4 x + 1) - 2 (4 x + 1)}{8 x^{3} + 2 x^{2} + 4 x + 1}$

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$\frac{(4 x + 1) (x - 2)}{8 x^{3} + 2 x^{2} + 4 x + 1}$

Step 2

Simplify the denominator.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(4 x + 1) (x - 2)}{(8 x^{3} + 2 x^{2}) + 4 x + 1}$

Factor out the greatest common factor (GCF) from each group.

$\frac{(4 x + 1) (x - 2)}{2 x^{2} (4 x + 1) + 1 (4 x + 1)}$

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$\frac{(4 x + 1) (x - 2)}{(4 x + 1) (2 x^{2} + 1)}$

Step 3

Cancel the common factor of $4 x + 1$ .

Cancel the common factor.

$\frac{(4 x + 1) (x - 2)}{(4 x + 1) (2 x^{2} + 1)}$

Rewrite the expression.

$\frac{x - 2}{2 x^{2} + 1}$

Simplify (4x^2-7x-2)/(8x^3+2x^2+4x+1)

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