Basic Math Examples

$\frac{3 z^{2} - 7 z - 20}{z + 7} \div \frac{z^{2} + 14 z + 49}{3 z^{2} + 26 z + 35}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 z^{2} - 7 z - 20}{z + 7} \cdot \frac{3 z^{2} + 26 z + 35}{z^{2} + 14 z + 49}$

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 $a \cdot c = 3 \cdot - 20 = - 60$ and whose sum is $b = - 7$ .

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Step 2.1.1

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

$\frac{3 z^{2} - 7 (z) - 20}{z + 7} \cdot \frac{3 z^{2} + 26 z + 35}{z^{2} + 14 z + 49}$

Step 2.1.2

Rewrite $- 7$ as $5$ plus $- 12$

$\frac{3 z^{2} + (5 - 12) z - 20}{z + 7} \cdot \frac{3 z^{2} + 26 z + 35}{z^{2} + 14 z + 49}$

Step 2.1.3

Apply the distributive property.

$\frac{3 z^{2} + 5 z - 12 z - 20}{z + 7} \cdot \frac{3 z^{2} + 26 z + 35}{z^{2} + 14 z + 49}$

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.

$\frac{(3 z^{2} + 5 z) - 12 z - 20}{z + 7} \cdot \frac{3 z^{2} + 26 z + 35}{z^{2} + 14 z + 49}$

Step 2.2.2

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

$\frac{z (3 z + 5) - 4 (3 z + 5)}{z + 7} \cdot \frac{3 z^{2} + 26 z + 35}{z^{2} + 14 z + 49}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $3 z + 5$ .

$\frac{(3 z + 5) (z - 4)}{z + 7} \cdot \frac{3 z^{2} + 26 z + 35}{z^{2} + 14 z + 49}$

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 $a \cdot c = 3 \cdot 35 = 105$ and whose sum is $b = 26$ .

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Step 3.1.1

Factor $26$ out of $26 z$ .

$\frac{(3 z + 5) (z - 4)}{z + 7} \cdot \frac{3 z^{2} + 26 (z) + 35}{z^{2} + 14 z + 49}$

Step 3.1.2

Rewrite $26$ as $5$ plus $21$

$\frac{(3 z + 5) (z - 4)}{z + 7} \cdot \frac{3 z^{2} + (5 + 21) z + 35}{z^{2} + 14 z + 49}$

Step 3.1.3

Apply the distributive property.

$\frac{(3 z + 5) (z - 4)}{z + 7} \cdot \frac{3 z^{2} + 5 z + 21 z + 35}{z^{2} + 14 z + 49}$

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.

$\frac{(3 z + 5) (z - 4)}{z + 7} \cdot \frac{(3 z^{2} + 5 z) + 21 z + 35}{z^{2} + 14 z + 49}$

Step 3.2.2

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

$\frac{(3 z + 5) (z - 4)}{z + 7} \cdot \frac{z (3 z + 5) + 7 (3 z + 5)}{z^{2} + 14 z + 49}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 z + 5$ .

$\frac{(3 z + 5) (z - 4)}{z + 7} \cdot \frac{(3 z + 5) (z + 7)}{z^{2} + 14 z + 49}$

Step 4

Factor using the perfect square rule.

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Step 4.1

Rewrite $49$ as $7^{2}$ .

$\frac{(3 z + 5) (z - 4)}{z + 7} \cdot \frac{(3 z + 5) (z + 7)}{z^{2} + 14 z + 7^{2}}$

Step 4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 z = 2 \cdot z \cdot 7$

Step 4.3

Rewrite the polynomial.

$\frac{(3 z + 5) (z - 4)}{z + 7} \cdot \frac{(3 z + 5) (z + 7)}{z^{2} + 2 \cdot z \cdot 7 + 7^{2}}$

Step 4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = z$ and $b = 7$ .

$\frac{(3 z + 5) (z - 4)}{z + 7} \cdot \frac{(3 z + 5) (z + 7)}{{(z + 7)}^{2}}$

Step 5

Cancel the common factor of $z + 7$ .

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Step 5.1

Factor $z + 7$ out of $(3 z + 5) (z + 7)$ .

$\frac{(3 z + 5) (z - 4)}{z + 7} \cdot \frac{(z + 7) (3 z + 5)}{{(z + 7)}^{2}}$

Step 5.2

Cancel the common factor.

$\frac{(3 z + 5) (z - 4)}{z + 7} \cdot \frac{(z + 7) (3 z + 5)}{{(z + 7)}^{2}}$

Step 5.3

Rewrite the expression.

$(3 z + 5) (z - 4) \frac{3 z + 5}{{(z + 7)}^{2}}$

Step 6

Expand $(3 z + 5) (z - 4)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$(3 z (z - 4) + 5 (z - 4)) \frac{3 z + 5}{{(z + 7)}^{2}}$

Step 6.2

Apply the distributive property.

$(3 z \cdot z + 3 z \cdot - 4 + 5 (z - 4)) \frac{3 z + 5}{{(z + 7)}^{2}}$

Step 6.3

Apply the distributive property.

$(3 z \cdot z + 3 z \cdot - 4 + 5 z + 5 \cdot - 4) \frac{3 z + 5}{{(z + 7)}^{2}}$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $z$ by $z$ by adding the exponents.

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Step 7.1.1.1

Move $z$ .

$(3 (z \cdot z) + 3 z \cdot - 4 + 5 z + 5 \cdot - 4) \frac{3 z + 5}{{(z + 7)}^{2}}$

Step 7.1.1.2

Multiply $z$ by $z$ .

$(3 z^{2} + 3 z \cdot - 4 + 5 z + 5 \cdot - 4) \frac{3 z + 5}{{(z + 7)}^{2}}$

Step 7.1.2

Multiply $- 4$ by $3$ .

$(3 z^{2} - 12 z + 5 z + 5 \cdot - 4) \frac{3 z + 5}{{(z + 7)}^{2}}$

Step 7.1.3

Multiply $5$ by $- 4$ .

$(3 z^{2} - 12 z + 5 z - 20) \frac{3 z + 5}{{(z + 7)}^{2}}$

Step 7.2

Add $- 12 z$ and $5 z$ .

$(3 z^{2} - 7 z - 20) \frac{3 z + 5}{{(z + 7)}^{2}}$

Step 8

Multiply $3 z^{2} - 7 z - 20$ by $\frac{3 z + 5}{{(z + 7)}^{2}}$ .

$\frac{(3 z^{2} - 7 z - 20) (3 z + 5)}{{(z + 7)}^{2}}$

Step 9

Simplify the numerator.

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Step 9.1

Factor by grouping.

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Step 9.1.1

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 $a \cdot c = 3 \cdot - 20 = - 60$ and whose sum is $b = - 7$ .

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Step 9.1.1.1

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

$\frac{(3 z^{2} - 7 (z) - 20) (3 z + 5)}{{(z + 7)}^{2}}$

Step 9.1.1.2

Rewrite $- 7$ as $5$ plus $- 12$

$\frac{(3 z^{2} + (5 - 12) z - 20) (3 z + 5)}{{(z + 7)}^{2}}$

Step 9.1.1.3

Apply the distributive property.

$\frac{(3 z^{2} + 5 z - 12 z - 20) (3 z + 5)}{{(z + 7)}^{2}}$

Step 9.1.2

Factor out the greatest common factor from each group.

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Step 9.1.2.1

Group the first two terms and the last two terms.

$\frac{((3 z^{2} + 5 z) - 12 z - 20) (3 z + 5)}{{(z + 7)}^{2}}$

Step 9.1.2.2

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

$\frac{(z (3 z + 5) - 4 (3 z + 5)) (3 z + 5)}{{(z + 7)}^{2}}$

Step 9.1.3

Factor the polynomial by factoring out the greatest common factor, $3 z + 5$ .

$\frac{(3 z + 5) (z - 4) (3 z + 5)}{{(z + 7)}^{2}}$

Step 9.2

Combine exponents.

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Step 9.2.1

Raise $3 z + 5$ to the power of $1$ .

$\frac{{(3 z + 5)}^{1} (3 z + 5) (z - 4)}{{(z + 7)}^{2}}$

Step 9.2.2

Raise $3 z + 5$ to the power of $1$ .

$\frac{{(3 z + 5)}^{1} {(3 z + 5)}^{1} (z - 4)}{{(z + 7)}^{2}}$

Step 9.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(3 z + 5)}^{1 + 1} (z - 4)}{{(z + 7)}^{2}}$

Step 9.2.4

Add $1$ and $1$ .

$\frac{{(3 z + 5)}^{2} (z - 4)}{{(z + 7)}^{2}}$