Basic Math Examples

$\frac{u^{2} - y^{2}}{2 u^{2} + 2 u y + u + y} \cdot \frac{2 u^{2} - 11 u - 6}{y u - 6 y - u^{2} + 6 u}$

Step 1

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

$\frac{(u + y) (u - y)}{2 u^{2} + 2 u y + u + y} \cdot \frac{2 u^{2} - 11 u - 6}{y u - 6 y - u^{2} + 6 u}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.1.1

Group the first two terms and the last two terms.

$\frac{(u + y) (u - y)}{(2 u^{2} + 2 u y) + u + y} \cdot \frac{2 u^{2} - 11 u - 6}{y u - 6 y - u^{2} + 6 u}$

Step 2.1.2

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

$\frac{(u + y) (u - y)}{2 u (u + y) + 1 (u + y)} \cdot \frac{2 u^{2} - 11 u - 6}{y u - 6 y - u^{2} + 6 u}$

Step 2.2

Factor the polynomial by factoring out the greatest common factor, $u + y$ .

$\frac{(u + y) (u - y)}{(u + y) (2 u + 1)} \cdot \frac{2 u^{2} - 11 u - 6}{y u - 6 y - u^{2} + 6 u}$

Step 3

Factor by grouping.

Tap for more steps...

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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = - 11$ .

Tap for more steps...

Step 3.1.1

Factor $- 11$ out of $- 11 u$ .

$\frac{(u + y) (u - y)}{(u + y) (2 u + 1)} \cdot \frac{2 u^{2} - 11 (u) - 6}{y u - 6 y - u^{2} + 6 u}$

Step 3.1.2

Rewrite $- 11$ as $1$ plus $- 12$

$\frac{(u + y) (u - y)}{(u + y) (2 u + 1)} \cdot \frac{2 u^{2} + (1 - 12) u - 6}{y u - 6 y - u^{2} + 6 u}$

Step 3.1.3

Apply the distributive property.

$\frac{(u + y) (u - y)}{(u + y) (2 u + 1)} \cdot \frac{2 u^{2} + 1 u - 12 u - 6}{y u - 6 y - u^{2} + 6 u}$

Step 3.1.4

Multiply $u$ by $1$ .

$\frac{(u + y) (u - y)}{(u + y) (2 u + 1)} \cdot \frac{2 u^{2} + u - 12 u - 6}{y u - 6 y - u^{2} + 6 u}$

Step 3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.1

Group the first two terms and the last two terms.

$\frac{(u + y) (u - y)}{(u + y) (2 u + 1)} \cdot \frac{(2 u^{2} + u) - 12 u - 6}{y u - 6 y - u^{2} + 6 u}$

Step 3.2.2

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

$\frac{(u + y) (u - y)}{(u + y) (2 u + 1)} \cdot \frac{u (2 u + 1) - 6 (2 u + 1)}{y u - 6 y - u^{2} + 6 u}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$\frac{(u + y) (u - y)}{(u + y) (2 u + 1)} \cdot \frac{(2 u + 1) (u - 6)}{y u - 6 y - u^{2} + 6 u}$

Step 4

Simplify the denominator.

Tap for more steps...

Step 4.1

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.1.1

Group the first two terms and the last two terms.

$\frac{(u + y) (u - y)}{(u + y) (2 u + 1)} \cdot \frac{(2 u + 1) (u - 6)}{(y u - 6 y) - u^{2} + 6 u}$

Step 4.1.2

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

$\frac{(u + y) (u - y)}{(u + y) (2 u + 1)} \cdot \frac{(2 u + 1) (u - 6)}{y (u - 6) - u (u - 6)}$

Step 4.2

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

$\frac{(u + y) (u - y)}{(u + y) (2 u + 1)} \cdot \frac{(2 u + 1) (u - 6)}{(u - 6) (y - u)}$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Cancel the common factor of $2 u + 1$ .

Tap for more steps...

Step 5.1.1

Factor $2 u + 1$ out of $(u + y) (2 u + 1)$ .

$\frac{(u + y) (u - y)}{(2 u + 1) (u + y)} \cdot \frac{(2 u + 1) (u - 6)}{(u - 6) (y - u)}$

Step 5.1.2

Cancel the common factor.

$\frac{(u + y) (u - y)}{(2 u + 1) (u + y)} \cdot \frac{(2 u + 1) (u - 6)}{(u - 6) (y - u)}$

Step 5.1.3

Rewrite the expression.

$\frac{(u + y) (u - y)}{u + y} \cdot \frac{u - 6}{(u - 6) (y - u)}$

Step 5.2

Multiply $\frac{(u + y) (u - y)}{u + y}$ by $\frac{u - 6}{(u - 6) (y - u)}$ .

$\frac{(u + y) (u - y) (u - 6)}{(u + y) ((u - 6) (y - u))}$

Step 5.3

Cancel the common factor of $u + y$ .

Tap for more steps...

Step 5.3.1

Cancel the common factor.

$\frac{(u + y) (u - y) (u - 6)}{(u + y) ((u - 6) (y - u))}$

Step 5.3.2

Rewrite the expression.

$\frac{(u - y) (u - 6)}{(u - 6) (y - u)}$

Step 5.4

Cancel the common factor of $u - 6$ .

Tap for more steps...

Step 5.4.1

Cancel the common factor.

$\frac{(u - y) (u - 6)}{(u - 6) (y - u)}$

Step 5.4.2

Rewrite the expression.

$\frac{u - y}{y - u}$

Step 5.5

Cancel the common factor of $u - y$ and $y - u$ .

Tap for more steps...

Step 5.5.1

Factor $- 1$ out of $u$ .

$\frac{- 1 (- u) - y}{y - u}$

Step 5.5.2

Factor $- 1$ out of $- y$ .

$\frac{- 1 (- u) - (y)}{y - u}$

Step 5.5.3

Factor $- 1$ out of $- 1 (- u) - (y)$ .

$\frac{- 1 (- u + y)}{y - u}$

Step 5.5.4

Reorder terms.

$\frac{- 1 (- u + y)}{- u + y}$

Step 5.5.5

Cancel the common factor.

$\frac{- 1 (- u + y)}{- u + y}$

Step 5.5.6

Divide $- 1$ by $1$ .

$- 1$