Simplify (12z^2-7z-12)/(3z^2+2z-8)

$\frac{12 z^{2} - 7 z - 12}{3 z^{2} + 2 z - 8}$

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 $a \cdot c = 12 \cdot - 12 = - 144$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 z$ .

$\frac{12 z^{2} - 7 (z) - 12}{3 z^{2} + 2 z - 8}$

Rewrite $- 7$ as $9$ plus $- 16$

$\frac{12 z^{2} + (9 - 16) z - 12}{3 z^{2} + 2 z - 8}$

Apply the distributive property.

$\frac{12 z^{2} + 9 z - 16 z - 12}{3 z^{2} + 2 z - 8}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(12 z^{2} + 9 z) - 16 z - 12}{3 z^{2} + 2 z - 8}$

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

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

Factor the polynomial by factoring out the greatest common factor, $4 z + 3$ .

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

Step 2

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 $a \cdot c = 3 \cdot - 8 = - 24$ and whose sum is $b = 2$ .

Factor $2$ out of $2 z$ .

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

Rewrite $2$ as $- 4$ plus $6$

$\frac{(4 z + 3) (3 z - 4)}{3 z^{2} + (- 4 + 6) z - 8}$

Apply the distributive property.

$\frac{(4 z + 3) (3 z - 4)}{3 z^{2} - 4 z + 6 z - 8}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(4 z + 3) (3 z - 4)}{(3 z^{2} - 4 z) + 6 z - 8}$

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

$\frac{(4 z + 3) (3 z - 4)}{z (3 z - 4) + 2 (3 z - 4)}$

Factor the polynomial by factoring out the greatest common factor, $3 z - 4$ .

$\frac{(4 z + 3) (3 z - 4)}{(3 z - 4) (z + 2)}$

Step 3

Cancel the common factor of $3 z - 4$ .

Cancel the common factor.

$\frac{(4 z + 3) (3 z - 4)}{(3 z - 4) (z + 2)}$

Rewrite the expression.

$\frac{4 z + 3}{z + 2}$

Simplify (12z^2-7z-12)/(3z^2+2z-8)

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