Trigonometry Examples

$\frac{h - 2}{h^{2} + 4 h + 4} + \frac{h - 2}{h + 2}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $4$ as $2^{2}$ .

$\frac{h - 2}{h^{2} + 4 h + 2^{2}} + \frac{h - 2}{h + 2}$

Step 1.2

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

$4 h = 2 \cdot h \cdot 2$

Step 1.3

Rewrite the polynomial.

$\frac{h - 2}{h^{2} + 2 \cdot h \cdot 2 + 2^{2}} + \frac{h - 2}{h + 2}$

Step 1.4

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

$\frac{h - 2}{{(h + 2)}^{2}} + \frac{h - 2}{h + 2}$

Step 2

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

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

Step 3

Write each expression with a common denominator of ${(h + 2)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{h - 2}{h + 2}$ by $\frac{h + 2}{h + 2}$ .

$\frac{h - 2}{{(h + 2)}^{2}} + \frac{(h - 2) (h + 2)}{(h + 2) (h + 2)}$

Step 3.2

Raise $h + 2$ to the power of $1$ .

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

Step 3.3

Raise $h + 2$ to the power of $1$ .

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

$\frac{h - 2}{{(h + 2)}^{2}} + \frac{(h - 2) (h + 2)}{{(h + 2)}^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{h - 2 + (h - 2) (h + 2)}{{(h + 2)}^{2}}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(h - 2) (h + 2)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{h - 2 + h (h + 2) - 2 (h + 2)}{{(h + 2)}^{2}}$

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Combine the opposite terms in $h \cdot h + h \cdot 2 - 2 h - 2 \cdot 2$ .

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Step 5.2.1

Reorder the factors in the terms $h \cdot 2$ and $- 2 h$ .

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

Step 5.2.2

Subtract $2 h$ from $2 h$ .

$\frac{h - 2 + h \cdot h + 0 - 2 \cdot 2}{{(h + 2)}^{2}}$

Step 5.2.3

Add $h \cdot h$ and $0$ .

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

Step 5.3

Simplify each term.

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Step 5.3.1

Multiply $h$ by $h$ .

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

Step 5.3.2

Multiply $- 2$ by $2$ .

$\frac{h - 2 + h^{2} - 4}{{(h + 2)}^{2}}$

Step 5.4

Subtract $4$ from $- 2$ .

$\frac{h + h^{2} - 6}{{(h + 2)}^{2}}$

Step 5.5

Reorder terms.

$\frac{h^{2} + h - 6}{{(h + 2)}^{2}}$

Step 5.6

Factor $h^{2} + h - 6$ using the AC method.

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Step 5.6.1

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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 5.6.2

Write the factored form using these integers.

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