Basic Math Examples

$\frac{5 k^{2} - 4 k - 1}{5 k^{2} + 16 k + 11} \div \frac{5 k^{2} + 6 k - 11}{25 k^{2} - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{5 k^{2} - 4 k - 1}{5 k^{2} + 16 k + 11} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

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 $a \cdot c = 5 \cdot - 1 = - 5$ and whose sum is $b = - 4$ .

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Step 2.1.1

Factor $- 4$ out of $- 4 k$ .

$\frac{5 k^{2} - 4 (k) - 1}{5 k^{2} + 16 k + 11} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

Step 2.1.2

Rewrite $- 4$ as $1$ plus $- 5$

$\frac{5 k^{2} + (1 - 5) k - 1}{5 k^{2} + 16 k + 11} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

Step 2.1.3

Apply the distributive property.

$\frac{5 k^{2} + 1 k - 5 k - 1}{5 k^{2} + 16 k + 11} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

Step 2.1.4

Multiply $k$ by $1$ .

$\frac{5 k^{2} + k - 5 k - 1}{5 k^{2} + 16 k + 11} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

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.

$\frac{(5 k^{2} + k) - 5 k - 1}{5 k^{2} + 16 k + 11} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

Step 2.2.2

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

$\frac{k (5 k + 1) - (5 k + 1)}{5 k^{2} + 16 k + 11} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $5 k + 1$ .

$\frac{(5 k + 1) (k - 1)}{5 k^{2} + 16 k + 11} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

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 $a \cdot c = 5 \cdot 11 = 55$ and whose sum is $b = 16$ .

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Step 3.1.1

Factor $16$ out of $16 k$ .

$\frac{(5 k + 1) (k - 1)}{5 k^{2} + 16 (k) + 11} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

Step 3.1.2

Rewrite $16$ as $5$ plus $11$

$\frac{(5 k + 1) (k - 1)}{5 k^{2} + (5 + 11) k + 11} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

Step 3.1.3

Apply the distributive property.

$\frac{(5 k + 1) (k - 1)}{5 k^{2} + 5 k + 11 k + 11} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

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.

$\frac{(5 k + 1) (k - 1)}{(5 k^{2} + 5 k) + 11 k + 11} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

Step 3.2.2

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

$\frac{(5 k + 1) (k - 1)}{5 k (k + 1) + 11 (k + 1)} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

Step 3.3

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

$\frac{(5 k + 1) (k - 1)}{(k + 1) (5 k + 11)} \cdot \frac{25 k^{2} - 1}{5 k^{2} + 6 k - 11}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $25 k^{2}$ as ${(5 k)}^{2}$ .

$\frac{(5 k + 1) (k - 1)}{(k + 1) (5 k + 11)} \cdot \frac{{(5 k)}^{2} - 1}{5 k^{2} + 6 k - 11}$

Step 4.2

Rewrite $1$ as $1^{2}$ .

$\frac{(5 k + 1) (k - 1)}{(k + 1) (5 k + 11)} \cdot \frac{{(5 k)}^{2} - 1^{2}}{5 k^{2} + 6 k - 11}$

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 k$ and $b = 1$ .

$\frac{(5 k + 1) (k - 1)}{(k + 1) (5 k + 11)} \cdot \frac{(5 k + 1) (5 k - 1)}{5 k^{2} + 6 k - 11}$

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 $a \cdot c = 5 \cdot - 11 = - 55$ and whose sum is $b = 6$ .

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Step 5.1.1

Factor $6$ out of $6 k$ .

$\frac{(5 k + 1) (k - 1)}{(k + 1) (5 k + 11)} \cdot \frac{(5 k + 1) (5 k - 1)}{5 k^{2} + 6 (k) - 11}$

Step 5.1.2

Rewrite $6$ as $- 5$ plus $11$

$\frac{(5 k + 1) (k - 1)}{(k + 1) (5 k + 11)} \cdot \frac{(5 k + 1) (5 k - 1)}{5 k^{2} + (- 5 + 11) k - 11}$

Step 5.1.3

Apply the distributive property.

$\frac{(5 k + 1) (k - 1)}{(k + 1) (5 k + 11)} \cdot \frac{(5 k + 1) (5 k - 1)}{5 k^{2} - 5 k + 11 k - 11}$

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.

$\frac{(5 k + 1) (k - 1)}{(k + 1) (5 k + 11)} \cdot \frac{(5 k + 1) (5 k - 1)}{(5 k^{2} - 5 k) + 11 k - 11}$

Step 5.2.2

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

$\frac{(5 k + 1) (k - 1)}{(k + 1) (5 k + 11)} \cdot \frac{(5 k + 1) (5 k - 1)}{5 k (k - 1) + 11 (k - 1)}$

Step 5.3

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

$\frac{(5 k + 1) (k - 1)}{(k + 1) (5 k + 11)} \cdot \frac{(5 k + 1) (5 k - 1)}{(k - 1) (5 k + 11)}$

Step 6

Cancel the common factor of $k - 1$ .

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Step 6.1

Factor $k - 1$ out of $(5 k + 1) (k - 1)$ .

$\frac{(k - 1) (5 k + 1)}{(k + 1) (5 k + 11)} \cdot \frac{(5 k + 1) (5 k - 1)}{(k - 1) (5 k + 11)}$

Step 6.2

Cancel the common factor.

$\frac{(k - 1) (5 k + 1)}{(k + 1) (5 k + 11)} \cdot \frac{(5 k + 1) (5 k - 1)}{(k - 1) (5 k + 11)}$

Step 6.3

Rewrite the expression.

$\frac{5 k + 1}{(k + 1) (5 k + 11)} \cdot \frac{(5 k + 1) (5 k - 1)}{5 k + 11}$

Step 7

Multiply $\frac{5 k + 1}{(k + 1) (5 k + 11)}$ by $\frac{(5 k + 1) (5 k - 1)}{5 k + 11}$ .

$\frac{(5 k + 1) ((5 k + 1) (5 k - 1))}{(k + 1) (5 k + 11) (5 k + 11)}$

Step 8

Raise $5 k + 1$ to the power of $1$ .

$\frac{{(5 k + 1)}^{1} (5 k + 1) (5 k - 1)}{(k + 1) (5 k + 11) (5 k + 11)}$

Step 9

Raise $5 k + 1$ to the power of $1$ .

$\frac{{(5 k + 1)}^{1} {(5 k + 1)}^{1} (5 k - 1)}{(k + 1) (5 k + 11) (5 k + 11)}$

Step 10

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

$\frac{{(5 k + 1)}^{1 + 1} (5 k - 1)}{(k + 1) (5 k + 11) (5 k + 11)}$

Step 11

Add $1$ and $1$ .

$\frac{{(5 k + 1)}^{2} (5 k - 1)}{(k + 1) (5 k + 11) (5 k + 11)}$

Step 12

Raise $5 k + 11$ to the power of $1$ .

$\frac{{(5 k + 1)}^{2} (5 k - 1)}{(k + 1) ({(5 k + 11)}^{1} (5 k + 11))}$

Step 13

Raise $5 k + 11$ to the power of $1$ .

$\frac{{(5 k + 1)}^{2} (5 k - 1)}{(k + 1) ({(5 k + 11)}^{1} {(5 k + 11)}^{1})}$

Step 14

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

$\frac{{(5 k + 1)}^{2} (5 k - 1)}{(k + 1) {(5 k + 11)}^{1 + 1}}$

Step 15

Add $1$ and $1$ .

$\frac{{(5 k + 1)}^{2} (5 k - 1)}{(k + 1) {(5 k + 11)}^{2}}$