Simplify (3z^2-13z+4)/(z^4-2z^3-8z^2)

$\frac{3 z^{2} - 13 z + 4}{z^{4} - 2 z^{3} - 8 z^{2}}$

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 $a \cdot c = 3 \cdot 4 = 12$ and whose sum is $b = - 13$ .

Factor $- 13$ out of $- 13 z$ .

$\frac{3 z^{2} - 13 (z) + 4}{z^{4} - 2 z^{3} - 8 z^{2}}$

Rewrite $- 13$ as $- 1$ plus $- 12$

$\frac{3 z^{2} + (- 1 - 12) z + 4}{z^{4} - 2 z^{3} - 8 z^{2}}$

Apply the distributive property.

$\frac{3 z^{2} - 1 z - 12 z + 4}{z^{4} - 2 z^{3} - 8 z^{2}}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(3 z^{2} - 1 z) - 12 z + 4}{z^{4} - 2 z^{3} - 8 z^{2}}$

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

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

Factor the polynomial by factoring out the greatest common factor, $3 z - 1$ .

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

Step 2

Simplify the denominator.

Factor $z^{2}$ out of $z^{4} - 2 z^{3} - 8 z^{2}$ .

Factor $z^{2}$ out of $z^{4}$ .

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

Factor $z^{2}$ out of $- 2 z^{3}$ .

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

Factor $z^{2}$ out of $- 8 z^{2}$ .

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

Factor $z^{2}$ out of $z^{2} z^{2} + z^{2} (- 2 z)$ .

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

Factor $z^{2}$ out of $z^{2} (z^{2} - 2 z) + z^{2} \cdot - 8$ .

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

Factor $z^{2} - 2 z - 8$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Write the factored form using these integers.

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

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

Step 3

Cancel the common factor of $z - 4$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{3 z - 1}{z^{2} (z + 2)}$

Simplify (3z^2-13z+4)/(z^4-2z^3-8z^2)

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