Pre-Algebra Examples

$\frac{6 z^{2} - 53 z + 40}{6 z^{2} + 31 z - 30} \div \frac{48 z^{2} - 70 z + 25}{8 z^{2} + 43 z - 30}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{6 z^{2} - 53 z + 40}{6 z^{2} + 31 z - 30} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 6 \cdot 40 = 240$ and whose sum is $b = - 53$ .

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Step 2.1.1

Factor $- 53$ out of $- 53 z$ .

$\frac{6 z^{2} - 53 (z) + 40}{6 z^{2} + 31 z - 30} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 2.1.2

Rewrite $- 53$ as $- 5$ plus $- 48$

$\frac{6 z^{2} + (- 5 - 48) z + 40}{6 z^{2} + 31 z - 30} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 2.1.3

Apply the distributive property.

$\frac{6 z^{2} - 5 z - 48 z + 40}{6 z^{2} + 31 z - 30} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(6 z^{2} - 5 z) - 48 z + 40}{6 z^{2} + 31 z - 30} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 2.2.2

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

$\frac{z (6 z - 5) - 8 (6 z - 5)}{6 z^{2} + 31 z - 30} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $6 z - 5$ .

$\frac{(6 z - 5) (z - 8)}{6 z^{2} + 31 z - 30} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 6 \cdot - 30 = - 180$ and whose sum is $b = 31$ .

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Step 3.1.1

Factor $31$ out of $31 z$ .

$\frac{(6 z - 5) (z - 8)}{6 z^{2} + 31 (z) - 30} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 3.1.2

Rewrite $31$ as $- 5$ plus $36$

$\frac{(6 z - 5) (z - 8)}{6 z^{2} + (- 5 + 36) z - 30} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 3.1.3

Apply the distributive property.

$\frac{(6 z - 5) (z - 8)}{6 z^{2} - 5 z + 36 z - 30} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{(6 z - 5) (z - 8)}{(6 z^{2} - 5 z) + 36 z - 30} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 3.2.2

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

$\frac{(6 z - 5) (z - 8)}{z (6 z - 5) + 6 (6 z - 5)} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $6 z - 5$ .

$\frac{(6 z - 5) (z - 8)}{(6 z - 5) (z + 6)} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 4

Cancel the common factor of $6 z - 5$ .

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Step 4.1

Cancel the common factor.

$\frac{(6 z - 5) (z - 8)}{(6 z - 5) (z + 6)} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 4.2

Rewrite the expression.

$\frac{z - 8}{z + 6} \cdot \frac{8 z^{2} + 43 z - 30}{48 z^{2} - 70 z + 25}$

Step 5

Factor by grouping.

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Step 5.1

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 $a \cdot c = 8 \cdot - 30 = - 240$ and whose sum is $b = 43$ .

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Step 5.1.1

Factor $43$ out of $43 z$ .

$\frac{z - 8}{z + 6} \cdot \frac{8 z^{2} + 43 (z) - 30}{48 z^{2} - 70 z + 25}$

Step 5.1.2

Rewrite $43$ as $- 5$ plus $48$

$\frac{z - 8}{z + 6} \cdot \frac{8 z^{2} + (- 5 + 48) z - 30}{48 z^{2} - 70 z + 25}$

Step 5.1.3

Apply the distributive property.

$\frac{z - 8}{z + 6} \cdot \frac{8 z^{2} - 5 z + 48 z - 30}{48 z^{2} - 70 z + 25}$

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

$\frac{z - 8}{z + 6} \cdot \frac{(8 z^{2} - 5 z) + 48 z - 30}{48 z^{2} - 70 z + 25}$

Step 5.2.2

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

$\frac{z - 8}{z + 6} \cdot \frac{z (8 z - 5) + 6 (8 z - 5)}{48 z^{2} - 70 z + 25}$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $8 z - 5$ .

$\frac{z - 8}{z + 6} \cdot \frac{(8 z - 5) (z + 6)}{48 z^{2} - 70 z + 25}$

Step 6

Factor by grouping.

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Step 6.1

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 $a \cdot c = 48 \cdot 25 = 1200$ and whose sum is $b = - 70$ .

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Step 6.1.1

Factor $- 70$ out of $- 70 z$ .

$\frac{z - 8}{z + 6} \cdot \frac{(8 z - 5) (z + 6)}{48 z^{2} - 70 (z) + 25}$

Step 6.1.2

Rewrite $- 70$ as $- 30$ plus $- 40$

$\frac{z - 8}{z + 6} \cdot \frac{(8 z - 5) (z + 6)}{48 z^{2} + (- 30 - 40) z + 25}$

Step 6.1.3

Apply the distributive property.

$\frac{z - 8}{z + 6} \cdot \frac{(8 z - 5) (z + 6)}{48 z^{2} - 30 z - 40 z + 25}$

Step 6.2

Factor out the greatest common factor from each group.

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Step 6.2.1

Group the first two terms and the last two terms.

$\frac{z - 8}{z + 6} \cdot \frac{(8 z - 5) (z + 6)}{(48 z^{2} - 30 z) - 40 z + 25}$

Step 6.2.2

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

$\frac{z - 8}{z + 6} \cdot \frac{(8 z - 5) (z + 6)}{6 z (8 z - 5) - 5 (8 z - 5)}$

Step 6.3

Factor the polynomial by factoring out the greatest common factor, $8 z - 5$ .

$\frac{z - 8}{z + 6} \cdot \frac{(8 z - 5) (z + 6)}{(8 z - 5) (6 z - 5)}$

Step 7

Cancel the common factor of $z + 6$ .

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Step 7.1

Factor $z + 6$ out of $(8 z - 5) (z + 6)$ .

$\frac{z - 8}{z + 6} \cdot \frac{(z + 6) (8 z - 5)}{(8 z - 5) (6 z - 5)}$

Step 7.2

Cancel the common factor.

$\frac{z - 8}{z + 6} \cdot \frac{(z + 6) (8 z - 5)}{(8 z - 5) (6 z - 5)}$

Step 7.3

Rewrite the expression.

$(z - 8) \frac{8 z - 5}{(8 z - 5) (6 z - 5)}$

Step 8

Cancel the common factor of $8 z - 5$ .

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Step 8.1

Cancel the common factor.

$(z - 8) \frac{8 z - 5}{(8 z - 5) (6 z - 5)}$

Step 8.2

Rewrite the expression.

$(z - 8) \frac{1}{6 z - 5}$

Step 9

Multiply $z - 8$ by $\frac{1}{6 z - 5}$ .

$\frac{z - 8}{6 z - 5}$