Basic Math Examples

$\frac{y}{y + 1} + \frac{1}{y^{2} + 3 y + 2}$

Step 1

Factor $y^{2} + 3 y + 2$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 1.2

Write the factored form using these integers.

$\frac{y}{y + 1} + \frac{1}{(y + 1) (y + 2)}$

Step 2

To write $\frac{y}{y + 1}$ as a fraction with a common denominator, multiply by $\frac{y + 2}{y + 2}$ .

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{y}{y + 1}$ by $\frac{y + 2}{y + 2}$ .

$\frac{y (y + 2)}{(y + 1) (y + 2)} + \frac{1}{(y + 1) (y + 2)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{y (y + 2) + 1}{(y + 1) (y + 2)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $y$ by $y$ .

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

Step 4.3

Move $2$ to the left of $y$ .

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

Step 4.4

Factor using the perfect square rule.

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Step 4.4.1

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

$\frac{y^{2} + 2 y + 1^{2}}{(y + 1) (y + 2)}$

Step 4.4.2

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

$2 y = 2 \cdot y \cdot 1$

Step 4.4.3

Rewrite the polynomial.

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

Step 4.4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 1$ .

$\frac{{(y + 1)}^{2}}{(y + 1) (y + 2)}$

Step 5

Cancel the common factor of ${(y + 1)}^{2}$ and $y + 1$ .

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Step 5.1

Factor $y + 1$ out of ${(y + 1)}^{2}$ .

$\frac{(y + 1) (y + 1)}{(y + 1) (y + 2)}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Cancel the common factor.

$\frac{(y + 1) (y + 1)}{(y + 1) (y + 2)}$

Step 5.2.2

Rewrite the expression.

$\frac{y + 1}{y + 2}$