Pre-Algebra Examples

$\frac{s}{s^{2} + 6 s + 9} + \frac{5}{s^{2} + 4 s + 3}$

Step 1

Simplify each term.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

Rewrite $9$ as $3^{2}$ .

$\frac{s}{s^{2} + 6 s + 3^{2}} + \frac{5}{s^{2} + 4 s + 3}$

Step 1.1.2

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

$6 s = 2 \cdot s \cdot 3$

Step 1.1.3

Rewrite the polynomial.

$\frac{s}{s^{2} + 2 \cdot s \cdot 3 + 3^{2}} + \frac{5}{s^{2} + 4 s + 3}$

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = s$ and $b = 3$ .

$\frac{s}{{(s + 3)}^{2}} + \frac{5}{s^{2} + 4 s + 3}$

Step 1.2

Factor $s^{2} + 4 s + 3$ using the AC method.

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Step 1.2.1

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 $3$ and whose sum is $4$ .

$1, 3$

Step 1.2.2

Write the factored form using these integers.

$\frac{s}{{(s + 3)}^{2}} + \frac{5}{(s + 1) (s + 3)}$

Step 2

To write $\frac{s}{{(s + 3)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{s + 1}{s + 1}$ .

$\frac{s}{{(s + 3)}^{2}} \cdot \frac{s + 1}{s + 1} + \frac{5}{(s + 1) (s + 3)}$

Step 3

To write $\frac{5}{(s + 1) (s + 3)}$ as a fraction with a common denominator, multiply by $\frac{s + 3}{s + 3}$ .

$\frac{s}{{(s + 3)}^{2}} \cdot \frac{s + 1}{s + 1} + \frac{5}{(s + 1) (s + 3)} \cdot \frac{s + 3}{s + 3}$

Step 4

Write each expression with a common denominator of $(s + 1) {(s + 3)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{s}{{(s + 3)}^{2}}$ by $\frac{s + 1}{s + 1}$ .

$\frac{s (s + 1)}{{(s + 3)}^{2} (s + 1)} + \frac{5}{(s + 1) (s + 3)} \cdot \frac{s + 3}{s + 3}$

Step 4.2

Multiply $\frac{5}{(s + 1) (s + 3)}$ by $\frac{s + 3}{s + 3}$ .

$\frac{s (s + 1)}{{(s + 3)}^{2} (s + 1)} + \frac{5 (s + 3)}{(s + 1) (s + 3) (s + 3)}$

Step 4.3

Raise $s + 3$ to the power of $1$ .

$\frac{s (s + 1)}{{(s + 3)}^{2} (s + 1)} + \frac{5 (s + 3)}{(s + 1) ({(s + 3)}^{1} (s + 3))}$

Step 4.4

Raise $s + 3$ to the power of $1$ .

$\frac{s (s + 1)}{{(s + 3)}^{2} (s + 1)} + \frac{5 (s + 3)}{(s + 1) ({(s + 3)}^{1} {(s + 3)}^{1})}$

Step 4.5

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

$\frac{s (s + 1)}{{(s + 3)}^{2} (s + 1)} + \frac{5 (s + 3)}{(s + 1) {(s + 3)}^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

$\frac{s (s + 1)}{{(s + 3)}^{2} (s + 1)} + \frac{5 (s + 3)}{(s + 1) {(s + 3)}^{2}}$

Step 4.7

Reorder the factors of $(s + 1) {(s + 3)}^{2}$ .

$\frac{s (s + 1)}{{(s + 3)}^{2} (s + 1)} + \frac{5 (s + 3)}{{(s + 3)}^{2} (s + 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{s (s + 1) + 5 (s + 3)}{{(s + 3)}^{2} (s + 1)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{s \cdot s + s \cdot 1 + 5 (s + 3)}{{(s + 3)}^{2} (s + 1)}$

Step 6.2

Multiply $s$ by $s$ .

$\frac{s^{2} + s \cdot 1 + 5 (s + 3)}{{(s + 3)}^{2} (s + 1)}$

Step 6.3

Multiply $s$ by $1$ .

$\frac{s^{2} + s + 5 (s + 3)}{{(s + 3)}^{2} (s + 1)}$

Step 6.4

Apply the distributive property.

$\frac{s^{2} + s + 5 s + 5 \cdot 3}{{(s + 3)}^{2} (s + 1)}$

Step 6.5

Multiply $5$ by $3$ .

$\frac{s^{2} + s + 5 s + 15}{{(s + 3)}^{2} (s + 1)}$

Step 6.6

Add $s$ and $5 s$ .

$\frac{s^{2} + 6 s + 15}{{(s + 3)}^{2} (s + 1)}$