Algebra Examples

$\frac{x^{3} + x^{2} y + x y^{2} + y^{3}}{x^{2} + 2 x y + y^{2}} + \frac{2 x y}{x + y}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify the numerator.

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Step 1.1.1.1

Factor out the greatest common factor from each group.

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Step 1.1.1.1.1

Group the first two terms and the last two terms.

$\frac{(x^{3} + x^{2} y) + x y^{2} + y^{3}}{x^{2} + 2 x y + y^{2}} + \frac{2 x y}{x + y}$

Step 1.1.1.1.2

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

$\frac{x^{2} (x + y) + y^{2} (x + y)}{x^{2} + 2 x y + y^{2}} + \frac{2 x y}{x + y}$

Step 1.1.1.2

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

$\frac{(x + y) (x^{2} + y^{2})}{x^{2} + 2 x y + y^{2}} + \frac{2 x y}{x + y}$

Step 1.1.2

Factor using the perfect square rule.

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Step 1.1.2.1

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

$2 x y = 2 \cdot x \cdot y$

Step 1.1.2.2

Rewrite the polynomial.

$\frac{(x + y) (x^{2} + y^{2})}{x^{2} + 2 \cdot x \cdot y + y^{2}} + \frac{2 x y}{x + y}$

Step 1.1.2.3

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

$\frac{(x + y) (x^{2} + y^{2})}{{(x + y)}^{2}} + \frac{2 x y}{x + y}$

Step 1.1.3

Cancel the common factors.

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Step 1.1.3.1

Factor $x + y$ out of ${(x + y)}^{2}$ .

$\frac{(x + y) (x^{2} + y^{2})}{(x + y) (x + y)} + \frac{2 x y}{x + y}$

Step 1.1.3.2

Cancel the common factor.

$\frac{(x + y) (x^{2} + y^{2})}{(x + y) (x + y)} + \frac{2 x y}{x + y}$

Step 1.1.3.3

Rewrite the expression.

$\frac{x^{2} + y^{2}}{x + y} + \frac{2 x y}{x + y}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{x^{2} + y^{2} + 2 x y}{x + y}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rearrange terms.

$\frac{x^{2} + 2 x y + y^{2}}{x + y}$

Step 2.2

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

$2 x y = 2 \cdot x \cdot y$

Step 2.3

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot y + y^{2}}{x + y}$

Step 2.4

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

$\frac{{(x + y)}^{2}}{x + y}$

Step 3

Cancel the common factor of ${(x + y)}^{2}$ and $x + y$ .

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Step 3.1

Factor $x + y$ out of ${(x + y)}^{2}$ .

$\frac{(x + y) (x + y)}{x + y}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Multiply by $1$ .

$\frac{(x + y) (x + y)}{(x + y) \cdot 1}$

Step 3.2.2

Cancel the common factor.

$\frac{(x + y) (x + y)}{(x + y) \cdot 1}$

Step 3.2.3

Rewrite the expression.

$\frac{x + y}{1}$

Step 3.2.4

Divide $x + y$ by $1$ .

$x + y$