Basic Math Examples

$\frac{\frac{3 k^{2} - 2 k - 1}{3 k^{2} + 14 k + 11}}{\frac{9 k^{2} - 1}{3 k^{2} + 8 k - 11}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 k^{2} - 2 k - 1}{3 k^{2} + 14 k + 11} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

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 $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = - 2$ .

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Step 2.1.1

Factor $- 2$ out of $- 2 k$ .

$\frac{3 k^{2} - 2 (k) - 1}{3 k^{2} + 14 k + 11} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

Step 2.1.2

Rewrite $- 2$ as $1$ plus $- 3$

$\frac{3 k^{2} + (1 - 3) k - 1}{3 k^{2} + 14 k + 11} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

Step 2.1.3

Apply the distributive property.

$\frac{3 k^{2} + 1 k - 3 k - 1}{3 k^{2} + 14 k + 11} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

Step 2.1.4

Multiply $k$ by $1$ .

$\frac{3 k^{2} + k - 3 k - 1}{3 k^{2} + 14 k + 11} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

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.

$\frac{(3 k^{2} + k) - 3 k - 1}{3 k^{2} + 14 k + 11} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

Step 2.2.2

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

$\frac{k (3 k + 1) - (3 k + 1)}{3 k^{2} + 14 k + 11} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $3 k + 1$ .

$\frac{(3 k + 1) (k - 1)}{3 k^{2} + 14 k + 11} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

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 $a \cdot c = 3 \cdot 11 = 33$ and whose sum is $b = 14$ .

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Step 3.1.1

Factor $14$ out of $14 k$ .

$\frac{(3 k + 1) (k - 1)}{3 k^{2} + 14 (k) + 11} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

Step 3.1.2

Rewrite $14$ as $3$ plus $11$

$\frac{(3 k + 1) (k - 1)}{3 k^{2} + (3 + 11) k + 11} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

Step 3.1.3

Apply the distributive property.

$\frac{(3 k + 1) (k - 1)}{3 k^{2} + 3 k + 11 k + 11} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

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.

$\frac{(3 k + 1) (k - 1)}{(3 k^{2} + 3 k) + 11 k + 11} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

Step 3.2.2

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

$\frac{(3 k + 1) (k - 1)}{3 k (k + 1) + 11 (k + 1)} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

Step 3.3

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

$\frac{(3 k + 1) (k - 1)}{(k + 1) (3 k + 11)} \cdot \frac{3 k^{2} + 8 k - 11}{9 k^{2} - 1}$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 3 \cdot - 11 = - 33$ and whose sum is $b = 8$ .

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Step 4.1.1

Factor $8$ out of $8 k$ .

$\frac{(3 k + 1) (k - 1)}{(k + 1) (3 k + 11)} \cdot \frac{3 k^{2} + 8 (k) - 11}{9 k^{2} - 1}$

Step 4.1.2

Rewrite $8$ as $- 3$ plus $11$

$\frac{(3 k + 1) (k - 1)}{(k + 1) (3 k + 11)} \cdot \frac{3 k^{2} + (- 3 + 11) k - 11}{9 k^{2} - 1}$

Step 4.1.3

Apply the distributive property.

$\frac{(3 k + 1) (k - 1)}{(k + 1) (3 k + 11)} \cdot \frac{3 k^{2} - 3 k + 11 k - 11}{9 k^{2} - 1}$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$\frac{(3 k + 1) (k - 1)}{(k + 1) (3 k + 11)} \cdot \frac{(3 k^{2} - 3 k) + 11 k - 11}{9 k^{2} - 1}$

Step 4.2.2

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

$\frac{(3 k + 1) (k - 1)}{(k + 1) (3 k + 11)} \cdot \frac{3 k (k - 1) + 11 (k - 1)}{9 k^{2} - 1}$

Step 4.3

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

$\frac{(3 k + 1) (k - 1)}{(k + 1) (3 k + 11)} \cdot \frac{(k - 1) (3 k + 11)}{9 k^{2} - 1}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $9 k^{2}$ as ${(3 k)}^{2}$ .

$\frac{(3 k + 1) (k - 1)}{(k + 1) (3 k + 11)} \cdot \frac{(k - 1) (3 k + 11)}{{(3 k)}^{2} - 1}$

Step 5.2

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

$\frac{(3 k + 1) (k - 1)}{(k + 1) (3 k + 11)} \cdot \frac{(k - 1) (3 k + 11)}{{(3 k)}^{2} - 1^{2}}$

Step 5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 k$ and $b = 1$ .

$\frac{(3 k + 1) (k - 1)}{(k + 1) (3 k + 11)} \cdot \frac{(k - 1) (3 k + 11)}{(3 k + 1) (3 k - 1)}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $3 k + 1$ .

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Step 6.1.1

Cancel the common factor.

$\frac{(3 k + 1) (k - 1)}{(k + 1) (3 k + 11)} \cdot \frac{(k - 1) (3 k + 11)}{(3 k + 1) (3 k - 1)}$

Step 6.1.2

Rewrite the expression.

$\frac{k - 1}{(k + 1) (3 k + 11)} \cdot \frac{(k - 1) (3 k + 11)}{3 k - 1}$

Step 6.2

Cancel the common factor of $3 k + 11$ .

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Step 6.2.1

Factor $3 k + 11$ out of $(k + 1) (3 k + 11)$ .

$\frac{k - 1}{(3 k + 11) (k + 1)} \cdot \frac{(k - 1) (3 k + 11)}{3 k - 1}$

Step 6.2.2

Factor $3 k + 11$ out of $(k - 1) (3 k + 11)$ .

$\frac{k - 1}{(3 k + 11) (k + 1)} \cdot \frac{(3 k + 11) (k - 1)}{3 k - 1}$

Step 6.2.3

Cancel the common factor.

$\frac{k - 1}{(3 k + 11) (k + 1)} \cdot \frac{(3 k + 11) (k - 1)}{3 k - 1}$

Step 6.2.4

Rewrite the expression.

$\frac{k - 1}{k + 1} \cdot \frac{k - 1}{3 k - 1}$

Step 6.3

Multiply $\frac{k - 1}{k + 1}$ by $\frac{k - 1}{3 k - 1}$ .

$\frac{(k - 1) (k - 1)}{(k + 1) (3 k - 1)}$

Step 7

Raise $k - 1$ to the power of $1$ .

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

Step 8

Raise $k - 1$ to the power of $1$ .

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

Step 9

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

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

Step 10

Add $1$ and $1$ .

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