Pre-Algebra Examples

$\frac{\frac{x^{2} + 4 x - 5}{x^{2} - 6 x + 9}}{\frac{1}{x + 5}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} + 4 x - 5}{x^{2} - 6 x + 9} (x + 5)$

Step 2

Factor $x^{2} + 4 x - 5$ using the AC method.

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Step 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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 1) (x + 5)}{x^{2} - 6 x + 9} (x + 5)$

Step 3

Factor using the perfect square rule.

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Step 3.1

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

$\frac{(x - 1) (x + 5)}{x^{2} - 6 x + 3^{2}} (x + 5)$

Step 3.2

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

$6 x = 2 \cdot x \cdot 3$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

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

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

Step 4

Multiply $\frac{(x - 1) (x + 5)}{{(x - 3)}^{2}}$ by $x + 5$ .

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

Step 5

Simplify the numerator.

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Step 5.1

Raise $x + 5$ to the power of $1$ .

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

Step 5.2

Raise $x + 5$ to the power of $1$ .

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 6

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

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

Step 7

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $x$ by $x$ .

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

Step 8.1.2

Move $5$ to the left of $x$ .

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

Step 8.1.3

Multiply $5$ by $5$ .

$\frac{(x - 1) (x^{2} + 5 x + 5 x + 25)}{{(x - 3)}^{2}}$

Step 8.2

Add $5 x$ and $5 x$ .

$\frac{(x - 1) (x^{2} + 10 x + 25)}{{(x - 3)}^{2}}$

Step 9

Expand $(x - 1) (x^{2} + 10 x + 25)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x \cdot x^{2} + x (10 x) + x \cdot 25 - 1 x^{2} - 1 (10 x) - 1 \cdot 25}{{(x - 3)}^{2}}$

Step 10

Simplify each term.

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Step 10.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 10.1.1

Multiply $x$ by $x^{2}$ .

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Step 10.1.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{2} + x (10 x) + x \cdot 25 - 1 x^{2} - 1 (10 x) - 1 \cdot 25}{{(x - 3)}^{2}}$

Step 10.1.1.2

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

$\frac{x^{1 + 2} + x (10 x) + x \cdot 25 - 1 x^{2} - 1 (10 x) - 1 \cdot 25}{{(x - 3)}^{2}}$

Step 10.1.2

Add $1$ and $2$ .

$\frac{x^{3} + x (10 x) + x \cdot 25 - 1 x^{2} - 1 (10 x) - 1 \cdot 25}{{(x - 3)}^{2}}$

Step 10.2

Rewrite using the commutative property of multiplication.

$\frac{x^{3} + 10 x \cdot x + x \cdot 25 - 1 x^{2} - 1 (10 x) - 1 \cdot 25}{{(x - 3)}^{2}}$

Step 10.3

Multiply $x$ by $x$ by adding the exponents.

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Step 10.3.1

Move $x$ .

$\frac{x^{3} + 10 (x \cdot x) + x \cdot 25 - 1 x^{2} - 1 (10 x) - 1 \cdot 25}{{(x - 3)}^{2}}$

Step 10.3.2

Multiply $x$ by $x$ .

$\frac{x^{3} + 10 x^{2} + x \cdot 25 - 1 x^{2} - 1 (10 x) - 1 \cdot 25}{{(x - 3)}^{2}}$

Step 10.4

Move $25$ to the left of $x$ .

$\frac{x^{3} + 10 x^{2} + 25 \cdot x - 1 x^{2} - 1 (10 x) - 1 \cdot 25}{{(x - 3)}^{2}}$

Step 10.5

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

$\frac{x^{3} + 10 x^{2} + 25 x - x^{2} - 1 (10 x) - 1 \cdot 25}{{(x - 3)}^{2}}$

Step 10.6

Multiply $10$ by $- 1$ .

$\frac{x^{3} + 10 x^{2} + 25 x - x^{2} - 10 x - 1 \cdot 25}{{(x - 3)}^{2}}$

Step 10.7

Multiply $- 1$ by $25$ .

$\frac{x^{3} + 10 x^{2} + 25 x - x^{2} - 10 x - 25}{{(x - 3)}^{2}}$

Step 11

Subtract $x^{2}$ from $10 x^{2}$ .

$\frac{x^{3} + 9 x^{2} + 25 x - 10 x - 25}{{(x - 3)}^{2}}$

Step 12

Subtract $10 x$ from $25 x$ .

$\frac{x^{3} + 9 x^{2} + 15 x - 25}{{(x - 3)}^{2}}$

Step 13

Split the fraction $\frac{x^{3} + 9 x^{2} + 15 x - 25}{{(x - 3)}^{2}}$ into two fractions.

$\frac{x^{3} + 9 x^{2} + 15 x}{{(x - 3)}^{2}} + \frac{- 25}{{(x - 3)}^{2}}$

Step 14

Split the fraction $\frac{x^{3} + 9 x^{2} + 15 x}{{(x - 3)}^{2}}$ into two fractions.

$\frac{x^{3} + 9 x^{2}}{{(x - 3)}^{2}} + \frac{15 x}{{(x - 3)}^{2}} + \frac{- 25}{{(x - 3)}^{2}}$

Step 15

Split the fraction $\frac{x^{3} + 9 x^{2}}{{(x - 3)}^{2}}$ into two fractions.

$\frac{x^{3}}{{(x - 3)}^{2}} + \frac{9 x^{2}}{{(x - 3)}^{2}} + \frac{15 x}{{(x - 3)}^{2}} + \frac{- 25}{{(x - 3)}^{2}}$

Step 16

Move the negative in front of the fraction.

$\frac{x^{3}}{{(x - 3)}^{2}} + \frac{9 x^{2}}{{(x - 3)}^{2}} + \frac{15 x}{{(x - 3)}^{2}} - \frac{25}{{(x - 3)}^{2}}$