Algebra Examples

$\frac{u^{3} + v^{3}}{u^{3} - v^{3}} \cdot \frac{u^{2} - v^{2}}{u^{2} + 2 u v + v^{2}}$

Step 1

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = u$ and $b = v$ .

$\frac{(u + v) (u^{2} - u v + v^{2})}{u^{3} - v^{3}} \cdot \frac{u^{2} - v^{2}}{u^{2} + 2 u v + v^{2}}$

Step 2

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

$\frac{(u + v) (u^{2} - u v + v^{2})}{(u - v) (u^{2} + u v + v^{2})} \cdot \frac{u^{2} - v^{2}}{u^{2} + 2 u v + v^{2}}$

Step 3

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 $b = v$ .

$\frac{(u + v) (u^{2} - u v + v^{2})}{(u - v) (u^{2} + u v + v^{2})} \cdot \frac{(u + v) (u - v)}{u^{2} + 2 u v + v^{2}}$

Step 4

Factor using the perfect square rule.

Tap for more steps...

Step 4.1

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

$2 u v = 2 \cdot u \cdot v$

Step 4.2

Rewrite the polynomial.

$\frac{(u + v) (u^{2} - u v + v^{2})}{(u - v) (u^{2} + u v + v^{2})} \cdot \frac{(u + v) (u - v)}{u^{2} + 2 \cdot u \cdot v + v^{2}}$

Step 4.3

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

$\frac{(u + v) (u^{2} - u v + v^{2})}{(u - v) (u^{2} + u v + v^{2})} \cdot \frac{(u + v) (u - v)}{{(u + v)}^{2}}$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Combine.

$\frac{(u + v) (u^{2} - u v + v^{2}) ((u + v) (u - v))}{(u - v) (u^{2} + u v + v^{2}) {(u + v)}^{2}}$

Step 5.2

Cancel the common factor of $u + v$ and ${(u + v)}^{2}$ .

Tap for more steps...

Step 5.2.1

Factor $u + v$ out of $(u + v) (u^{2} - u v + v^{2}) ((u + v) (u - v))$ .

$\frac{(u + v) ((u^{2} - u v + v^{2}) ((u + v) (u - v)))}{(u - v) (u^{2} + u v + v^{2}) {(u + v)}^{2}}$

Step 5.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.2.1

Factor $u + v$ out of $(u - v) (u^{2} + u v + v^{2}) {(u + v)}^{2}$ .

$\frac{(u + v) ((u^{2} - u v + v^{2}) ((u + v) (u - v)))}{(u + v) ((u - v) (u^{2} + u v + v^{2}) (u + v))}$

Step 5.2.2.2

Cancel the common factor.

$\frac{(u + v) ((u^{2} - u v + v^{2}) ((u + v) (u - v)))}{(u + v) ((u - v) (u^{2} + u v + v^{2}) (u + v))}$

Step 5.2.2.3

Rewrite the expression.

$\frac{(u^{2} - u v + v^{2}) ((u + v) (u - v))}{(u - v) (u^{2} + u v + v^{2}) (u + v)}$

Step 5.3

Cancel the common factor of $u + v$ .

Tap for more steps...

Step 5.3.1

Cancel the common factor.

$\frac{(u^{2} - u v + v^{2}) ((u + v) (u - v))}{(u - v) (u^{2} + u v + v^{2}) (u + v)}$

Step 5.3.2

Rewrite the expression.

$\frac{(u^{2} - u v + v^{2}) (u - v)}{(u - v) (u^{2} + u v + v^{2})}$

Step 5.4

Cancel the common factor of $u - v$ .

Tap for more steps...

Step 5.4.1

Cancel the common factor.

$\frac{(u^{2} - u v + v^{2}) (u - v)}{(u - v) (u^{2} + u v + v^{2})}$

Step 5.4.2

Rewrite the expression.

$\frac{u^{2} - u v + v^{2}}{u^{2} + u v + v^{2}}$