Algebra Examples

$\frac{2 k + 5}{2 k^{4} + 5 k^{3}} \div \frac{35 k^{2} + 2 k - 1}{7 k^{3} - k^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 k + 5}{2 k^{4} + 5 k^{3}} \cdot \frac{7 k^{3} - k^{2}}{35 k^{2} + 2 k - 1}$

Step 2

Factor $k^{3}$ out of $2 k^{4} + 5 k^{3}$ .

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Step 2.1

Factor $k^{3}$ out of $2 k^{4}$ .

$\frac{2 k + 5}{k^{3} (2 k) + 5 k^{3}} \cdot \frac{7 k^{3} - k^{2}}{35 k^{2} + 2 k - 1}$

Step 2.2

Factor $k^{3}$ out of $5 k^{3}$ .

$\frac{2 k + 5}{k^{3} (2 k) + k^{3} \cdot 5} \cdot \frac{7 k^{3} - k^{2}}{35 k^{2} + 2 k - 1}$

Step 2.3

Factor $k^{3}$ out of $k^{3} (2 k) + k^{3} \cdot 5$ .

$\frac{2 k + 5}{k^{3} (2 k + 5)} \cdot \frac{7 k^{3} - k^{2}}{35 k^{2} + 2 k - 1}$

Step 3

Cancel the common factor of $2 k + 5$ .

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Step 3.1

Cancel the common factor.

$\frac{2 k + 5}{k^{3} (2 k + 5)} \cdot \frac{7 k^{3} - k^{2}}{35 k^{2} + 2 k - 1}$

Step 3.2

Rewrite the expression.

$\frac{1}{k^{3}} \cdot \frac{7 k^{3} - k^{2}}{35 k^{2} + 2 k - 1}$

Step 4

Factor $k^{2}$ out of $7 k^{3} - k^{2}$ .

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Step 4.1

Factor $k^{2}$ out of $7 k^{3}$ .

$\frac{1}{k^{3}} \cdot \frac{k^{2} (7 k) - k^{2}}{35 k^{2} + 2 k - 1}$

Step 4.2

Factor $k^{2}$ out of $- k^{2}$ .

$\frac{1}{k^{3}} \cdot \frac{k^{2} (7 k) + k^{2} \cdot - 1}{35 k^{2} + 2 k - 1}$

Step 4.3

Factor $k^{2}$ out of $k^{2} (7 k) + k^{2} \cdot - 1$ .

$\frac{1}{k^{3}} \cdot \frac{k^{2} (7 k - 1)}{35 k^{2} + 2 k - 1}$

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 $a \cdot c = 35 \cdot - 1 = - 35$ and whose sum is $b = 2$ .

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Step 5.1.1

Factor $2$ out of $2 k$ .

$\frac{1}{k^{3}} \cdot \frac{k^{2} (7 k - 1)}{35 k^{2} + 2 (k) - 1}$

Step 5.1.2

Rewrite $2$ as $- 5$ plus $7$

$\frac{1}{k^{3}} \cdot \frac{k^{2} (7 k - 1)}{35 k^{2} + (- 5 + 7) k - 1}$

Step 5.1.3

Apply the distributive property.

$\frac{1}{k^{3}} \cdot \frac{k^{2} (7 k - 1)}{35 k^{2} - 5 k + 7 k - 1}$

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.

$\frac{1}{k^{3}} \cdot \frac{k^{2} (7 k - 1)}{(35 k^{2} - 5 k) + 7 k - 1}$

Step 5.2.2

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

$\frac{1}{k^{3}} \cdot \frac{k^{2} (7 k - 1)}{5 k (7 k - 1) + 1 (7 k - 1)}$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $7 k - 1$ .

$\frac{1}{k^{3}} \cdot \frac{k^{2} (7 k - 1)}{(7 k - 1) (5 k + 1)}$

Step 6

Combine.

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

Step 7

Cancel the common factor of $k^{2}$ and $k^{3}$ .

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Step 7.1

Factor $k^{2}$ out of $1 (k^{2} (7 k - 1))$ .

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $k^{2}$ out of $k^{3} ((7 k - 1) (5 k + 1))$ .

$\frac{k^{2} (1 (7 k - 1))}{k^{2} (k ((7 k - 1) (5 k + 1)))}$

Step 7.2.2

Cancel the common factor.

$\frac{k^{2} (1 (7 k - 1))}{k^{2} (k ((7 k - 1) (5 k + 1)))}$

Step 7.2.3

Rewrite the expression.

$\frac{1 (7 k - 1)}{k ((7 k - 1) (5 k + 1))}$

Step 8

Cancel the common factor of $7 k - 1$ .

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Step 8.1

Cancel the common factor.

$\frac{1 (7 k - 1)}{k ((7 k - 1) (5 k + 1))}$

Step 8.2

Rewrite the expression.

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