Pre-Algebra Examples

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

Step 1

Find where the expression $\frac{3 x^{2} (x + 4)}{(x + 3) (x - 5)}$ is undefined.

$x = - 3, x = 5$

Step 2

Since $\frac{3 x^{2} (x + 4)}{(x + 3) (x - 5)}$ $\to$ $\infty$ as $x$ $\to$ $- 3$ from the left and $\frac{3 x^{2} (x + 4)}{(x + 3) (x - 5)}$ $\to$ $- \infty$ as $x$ $\to$ $- 3$ from the right, then $x = - 3$ is a vertical asymptote.

$x = - 3$

Step 3

Since $\frac{3 x^{2} (x + 4)}{(x + 3) (x - 5)}$ $\to$ $- \infty$ as $x$ $\to$ $5$ from the left and $\frac{3 x^{2} (x + 4)}{(x + 3) (x - 5)}$ $\to$ $\infty$ as $x$ $\to$ $5$ from the right, then $x = 5$ is a vertical asymptote.

$x = 5$

Step 4

List all of the vertical asymptotes:

$x = - 3, 5$

Step 5

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 6

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 7

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 8

Find the oblique asymptote using polynomial division.

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Step 8.1

Simplify the expression.

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Step 8.1.1

Factor $3 x^{2}$ out of $3 x^{3} + 12 x^{2}$ .

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Step 8.1.1.1

Factor $3 x^{2}$ out of $3 x^{3}$ .

$\frac{3 x^{2} (x) + 12 x^{2}}{x^{2} - 2 x - 15}$

Step 8.1.1.2

Factor $3 x^{2}$ out of $12 x^{2}$ .

$\frac{3 x^{2} (x) + 3 x^{2} (4)}{x^{2} - 2 x - 15}$

Step 8.1.1.3

Factor $3 x^{2}$ out of $3 x^{2} (x) + 3 x^{2} (4)$ .

$\frac{3 x^{2} (x + 4)}{x^{2} - 2 x - 15}$

Step 8.1.2

Factor $x^{2} - 2 x - 15$ using the AC method.

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Step 8.1.2.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 8.1.2.2

Write the factored form using these integers.

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

Step 8.2

Expand $3 x^{2} (x + 4)$ .

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Step 8.2.1

Apply the distributive property.

$\frac{3 x^{2} x + 3 x^{2} \cdot 4}{(x - 5) (x + 3)}$

Step 8.2.2

Move $x^{2}$ .

$\frac{3 x^{2} x + 3 \cdot (4 x^{2})}{(x - 5) (x + 3)}$

Step 8.2.3

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

$\frac{3 (x^{2} x) + 3 \cdot (4 x^{2})}{(x - 5) (x + 3)}$

Step 8.2.4

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

$\frac{3 x^{2 + 1} + 3 \cdot (4 x^{2})}{(x - 5) (x + 3)}$

Step 8.2.5

Add $2$ and $1$ .

$\frac{3 x^{3} + 3 \cdot (4 x^{2})}{(x - 5) (x + 3)}$

Step 8.2.6

Multiply $3$ by $4$ .

$\frac{3 x^{3} + 12 x^{2}}{(x - 5) (x + 3)}$

Step 8.3

Expand $(x - 5) (x + 3)$ .

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Step 8.3.1

Apply the distributive property.

$\frac{3 x^{3} + 12 x^{2}}{x (x + 3) - 5 (x + 3)}$

Step 8.3.2

Apply the distributive property.

$\frac{3 x^{3} + 12 x^{2}}{x \cdot x + x \cdot 3 - 5 (x + 3)}$

Step 8.3.3

Apply the distributive property.

$\frac{3 x^{3} + 12 x^{2}}{x \cdot x + x \cdot 3 - 5 x - 5 \cdot 3}$

Step 8.3.4

Reorder $x$ and $3$ .

$\frac{3 x^{3} + 12 x^{2}}{x \cdot x + 3 x - 5 x - 5 \cdot 3}$

Step 8.3.5

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

$\frac{3 x^{3} + 12 x^{2}}{x \cdot x + 3 x - 5 x - 5 \cdot 3}$

Step 8.3.6

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

$\frac{3 x^{3} + 12 x^{2}}{x \cdot x + 3 x - 5 x - 5 \cdot 3}$

Step 8.3.7

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

$\frac{3 x^{3} + 12 x^{2}}{x^{1 + 1} + 3 x - 5 x - 5 \cdot 3}$

Step 8.3.8

Add $1$ and $1$ .

$\frac{3 x^{3} + 12 x^{2}}{x^{2} + 3 x - 5 x - 5 \cdot 3}$

Step 8.3.9

Multiply $- 5$ by $3$ .

$\frac{3 x^{3} + 12 x^{2}}{x^{2} + 3 x - 5 x - 15}$

Step 8.3.10

Subtract $5 x$ from $3 x$ .

$\frac{3 x^{3} + 12 x^{2}}{x^{2} - 2 x - 15}$

Step 8.4

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{2}$

$2 x$

$15$

$3 x^{3}$

$12 x^{2}$

$0 x$

$0$

Step 8.5

Divide the highest order term in the dividend $3 x^{3}$ by the highest order term in divisor $x^{2}$ .

$3 x$

$x^{2}$

$2 x$

$15$

$3 x^{3}$

$12 x^{2}$

$0 x$

$0$

Step 8.6

Multiply the new quotient term by the divisor.

$3 x$

$x^{2}$

$2 x$

$15$

$3 x^{3}$

$12 x^{2}$

$0 x$

$0$

$3 x^{3}$

$6 x^{2}$

$45 x$

Step 8.7

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} - 6 x^{2} - 45 x$

$3 x$

$x^{2}$

$2 x$

$15$

$3 x^{3}$

$12 x^{2}$

$0 x$

$0$

$3 x^{3}$

$6 x^{2}$

$45 x$

Step 8.8

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x$

$x^{2}$

$2 x$

$15$

$3 x^{3}$

$12 x^{2}$

$0 x$

$0$

$3 x^{3}$

$6 x^{2}$

$45 x$

$18 x^{2}$

$45 x$

Step 8.9

Pull the next terms from the original dividend down into the current dividend.

$3 x$

$x^{2}$

$2 x$

$15$

$3 x^{3}$

$12 x^{2}$

$0 x$

$0$

$3 x^{3}$

$6 x^{2}$

$45 x$

$18 x^{2}$

$45 x$

$0$

Step 8.10

Divide the highest order term in the dividend $18 x^{2}$ by the highest order term in divisor $x^{2}$ .

$3 x$

$18$

$x^{2}$

$2 x$

$15$

$3 x^{3}$

$12 x^{2}$

$0 x$

$0$

$3 x^{3}$

$6 x^{2}$

$45 x$

$18 x^{2}$

$45 x$

$0$

Step 8.11

Multiply the new quotient term by the divisor.

$3 x$

$18$

$x^{2}$

$2 x$

$15$

$3 x^{3}$

$12 x^{2}$

$0 x$

$0$

$3 x^{3}$

$6 x^{2}$

$45 x$

$18 x^{2}$

$45 x$

$0$

$18 x^{2}$

$36 x$

$270$

Step 8.12

The expression needs to be subtracted from the dividend, so change all the signs in $18 x^{2} - 36 x - 270$

$3 x$

$18$

$x^{2}$

$2 x$

$15$

$3 x^{3}$

$12 x^{2}$

$0 x$

$0$

$3 x^{3}$

$6 x^{2}$

$45 x$

$18 x^{2}$

$45 x$

$0$

$18 x^{2}$

$36 x$

$270$

Step 8.13

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$3 x$

$18$

$x^{2}$

$2 x$

$15$

$3 x^{3}$

$12 x^{2}$

$0 x$

$0$

$3 x^{3}$

$6 x^{2}$

$45 x$

$18 x^{2}$

$45 x$

$0$

$18 x^{2}$

$36 x$

$270$

$81 x$

$270$

Step 8.14

The final answer is the quotient plus the remainder over the divisor.

$3 x + 18 + \frac{81 x + 270}{x^{2} - 2 x - 15}$

Step 8.15

The oblique asymptote is the polynomial portion of the long division result.

$y = 3 x + 18$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 3, 5$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 3 x + 18$

Step 10