Pre-Algebra Examples

$\frac{3 x^{5} + 6 x^{4} + 54 x^{3} + 108 x^{2} - 525 x - 1050}{x^{2} + 25}$

Step 1

Find where the expression $\frac{3 (x^{5} + 2 x^{4} + 18 x^{3} + 36 x^{2} - 175 x - 350)}{x^{2} + 25}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

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 4

Find $n$ and $m$ .

$n = 5$

$m = 2$

Step 5

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

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

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Step 6.1

Factor $3$ out of $3 x^{5} + 6 x^{4} + 54 x^{3} + 108 x^{2} - 525 x - 1050$ .

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Step 6.1.1

Factor $3$ out of $3 x^{5}$ .

$\frac{3 (x^{5}) + 6 x^{4} + 54 x^{3} + 108 x^{2} - 525 x - 1050}{x^{2} + 25}$

Step 6.1.2

Factor $3$ out of $6 x^{4}$ .

$\frac{3 (x^{5}) + 3 (2 x^{4}) + 54 x^{3} + 108 x^{2} - 525 x - 1050}{x^{2} + 25}$

Step 6.1.3

Factor $3$ out of $54 x^{3}$ .

$\frac{3 (x^{5}) + 3 (2 x^{4}) + 3 (18 x^{3}) + 108 x^{2} - 525 x - 1050}{x^{2} + 25}$

Step 6.1.4

Factor $3$ out of $108 x^{2}$ .

$\frac{3 (x^{5}) + 3 (2 x^{4}) + 3 (18 x^{3}) + 3 (36 x^{2}) - 525 x - 1050}{x^{2} + 25}$

Step 6.1.5

Factor $3$ out of $- 525 x$ .

$\frac{3 (x^{5}) + 3 (2 x^{4}) + 3 (18 x^{3}) + 3 (36 x^{2}) + 3 (- 175 x) - 1050}{x^{2} + 25}$

Step 6.1.6

Factor $3$ out of $- 1050$ .

$\frac{3 x^{5} + 3 (2 x^{4}) + 3 (18 x^{3}) + 3 (36 x^{2}) + 3 (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.1.7

Factor $3$ out of $3 x^{5} + 3 (2 x^{4})$ .

$\frac{3 (x^{5} + 2 x^{4}) + 3 (18 x^{3}) + 3 (36 x^{2}) + 3 (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.1.8

Factor $3$ out of $3 (x^{5} + 2 x^{4}) + 3 (18 x^{3})$ .

$\frac{3 (x^{5} + 2 x^{4} + 18 x^{3}) + 3 (36 x^{2}) + 3 (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.1.9

Factor $3$ out of $3 (x^{5} + 2 x^{4} + 18 x^{3}) + 3 (36 x^{2})$ .

$\frac{3 (x^{5} + 2 x^{4} + 18 x^{3} + 36 x^{2}) + 3 (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.1.10

Factor $3$ out of $3 (x^{5} + 2 x^{4} + 18 x^{3} + 36 x^{2}) + 3 (- 175 x)$ .

$\frac{3 (x^{5} + 2 x^{4} + 18 x^{3} + 36 x^{2} - 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.1.11

Factor $3$ out of $3 (x^{5} + 2 x^{4} + 18 x^{3} + 36 x^{2} - 175 x) + 3 \cdot - 350$ .

$\frac{3 (x^{5} + 2 x^{4} + 18 x^{3} + 36 x^{2} - 175 x - 350)}{x^{2} + 25}$

$\frac{3 (x^{5} + 2 x^{4} + 18 x^{3} + 36 x^{2} - 175 x - 350)}{x^{2} + 25}$

Step 6.2

Expand $3 (x^{5} + 2 x^{4} + 18 x^{3} + 36 x^{2} - 175 x - 350)$ .

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Step 6.2.1

Apply the distributive property.

$\frac{3 (x^{5} + 2 x^{4} + 18 x^{3} + 36 x^{2} - 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.2

Apply the distributive property.

$\frac{3 (x^{5} + 2 x^{4} + 18 x^{3} + 36 x^{2}) + 3 (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.3

Apply the distributive property.

$\frac{3 (x^{5} + 2 x^{4} + 18 x^{3}) + 3 (36 x^{2}) + 3 (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.4

Apply the distributive property.

$\frac{3 (x^{5} + 2 x^{4}) + 3 (18 x^{3}) + 3 (36 x^{2}) + 3 (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.5

Apply the distributive property.

$\frac{3 x^{5} + 3 (2 x^{4}) + 3 (18 x^{3}) + 3 (36 x^{2}) + 3 (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.6

Remove parentheses.

$\frac{3 x^{5} + 3 \cdot (2 x^{4}) + 3 (18 x^{3}) + 3 (36 x^{2}) + 3 (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.7

Remove parentheses.

$\frac{3 x^{5} + 3 \cdot (2 x^{4}) + 3 \cdot (18 x^{3}) + 3 (36 x^{2}) + 3 (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.8

Remove parentheses.

$\frac{3 x^{5} + 3 \cdot (2 x^{4}) + 3 \cdot (18 x^{3}) + 3 \cdot (36 x^{2}) + 3 (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.9

Remove parentheses.

$\frac{3 x^{5} + 3 \cdot (2 x^{4}) + 3 \cdot (18 x^{3}) + 3 \cdot (36 x^{2}) + 3 \cdot (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.10

Multiply $3$ by $2$ .

$\frac{3 x^{5} + 6 x^{4} + 3 \cdot (18 x^{3}) + 3 \cdot (36 x^{2}) + 3 \cdot (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.11

Multiply $3$ by $18$ .

$\frac{3 x^{5} + 6 x^{4} + 54 x^{3} + 3 \cdot (36 x^{2}) + 3 \cdot (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.12

Multiply $3$ by $36$ .

$\frac{3 x^{5} + 6 x^{4} + 54 x^{3} + 108 x^{2} + 3 \cdot (- 175 x) + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.13

Multiply $3$ by $- 175$ .

$\frac{3 x^{5} + 6 x^{4} + 54 x^{3} + 108 x^{2} - 525 x + 3 \cdot - 350}{x^{2} + 25}$

Step 6.2.14

Multiply $3$ by $- 350$ .

$\frac{3 x^{5} + 6 x^{4} + 54 x^{3} + 108 x^{2} - 525 x - 1050}{x^{2} + 25}$

$\frac{3 x^{5} + 6 x^{4} + 54 x^{3} + 108 x^{2} - 525 x - 1050}{x^{2} + 25}$

Step 6.3

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

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

Step 6.4

Divide the highest order term in the dividend $3 x^{5}$ by the highest order term in divisor $x^{2}$ .

$3 x^{3}$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

Step 6.5

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

+

$3 x^{5}$

+

$0$

+

$75 x^{3}$

Step 6.6

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{5} + 0 + 75 x^{3}$

$3 x^{3}$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

-

$3 x^{5}$

-

$0$

-

$75 x^{3}$

Step 6.7

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

$3 x^{3}$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

-

$3 x^{5}$

-

$0$

-

$75 x^{3}$

+

$6 x^{4}$

-

$21 x^{3}$

Step 6.8

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

$3 x^{3}$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

-

$3 x^{5}$

-

$0$

-

$75 x^{3}$

+

$6 x^{4}$

-

$21 x^{3}$

+

$108 x^{2}$

Step 6.9

Divide the highest order term in the dividend $6 x^{4}$ by the highest order term in divisor $x^{2}$ .

$3 x^{3}$

+

$6 x^{2}$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

-

$3 x^{5}$

-

$0$

-

$75 x^{3}$

+

$6 x^{4}$

-

$21 x^{3}$

+

$108 x^{2}$

Step 6.10

Multiply the new quotient term by the divisor.

$3 x^{3}$

+

$6 x^{2}$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

-

$3 x^{5}$

-

$0$

-

$75 x^{3}$

+

$6 x^{4}$

-

$21 x^{3}$

+

$108 x^{2}$

+

$6 x^{4}$

+

$0$

+

$150 x^{2}$

Step 6.11

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{4} + 0 + 150 x^{2}$

$3 x^{3}$

+

$6 x^{2}$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

-

$3 x^{5}$

-

$0$

-

$75 x^{3}$

+

$6 x^{4}$

-

$21 x^{3}$

+

$108 x^{2}$

-

$6 x^{4}$

-

$0$

-

$150 x^{2}$

Step 6.12

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

$3 x^{3}$

+

$6 x^{2}$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

-

$3 x^{5}$

-

$0$

-

$75 x^{3}$

+

$6 x^{4}$

-

$21 x^{3}$

+

$108 x^{2}$

-

$6 x^{4}$

-

$0$

-

$150 x^{2}$

-

$21 x^{3}$

-

$42 x^{2}$

Step 6.13

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

$3 x^{3}$

+

$6 x^{2}$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

-

$3 x^{5}$

-

$0$

-

$75 x^{3}$

+

$6 x^{4}$

-

$21 x^{3}$

+

$108 x^{2}$

-

$6 x^{4}$

-

$0$

-

$150 x^{2}$

-

$21 x^{3}$

-

$42 x^{2}$

-

$525 x$

Step 6.14

Divide the highest order term in the dividend $- 21 x^{3}$ by the highest order term in divisor $x^{2}$ .

$3 x^{3}$

+

$6 x^{2}$

-

$21 x$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

-

$3 x^{5}$

-

$0$

-

$75 x^{3}$

+

$6 x^{4}$

-

$21 x^{3}$

+

$108 x^{2}$

-

$6 x^{4}$

-

$0$

-

$150 x^{2}$

-

$21 x^{3}$

-

$42 x^{2}$

-

$525 x$

Step 6.15

Multiply the new quotient term by the divisor.

$3 x^{3}$

+

$6 x^{2}$

-

$21 x$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

-

$3 x^{5}$

-

$0$

-

$75 x^{3}$

+

$6 x^{4}$

-

$21 x^{3}$

+

$108 x^{2}$

-

$6 x^{4}$

-

$0$

-

$150 x^{2}$

-

$21 x^{3}$

-

$42 x^{2}$

-

$525 x$

-

$21 x^{3}$

+

$0$

-

$525 x$

Step 6.16

The expression needs to be subtracted from the dividend, so change all the signs in $- 21 x^{3} + 0 - 525 x$

$3 x^{3}$

+

$6 x^{2}$

-

$21 x$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

-

$3 x^{5}$

-

$0$

-

$75 x^{3}$

+

$6 x^{4}$

-

$21 x^{3}$

+

$108 x^{2}$

-

$6 x^{4}$

-

$0$

-

$150 x^{2}$

-

$21 x^{3}$

-

$42 x^{2}$

-

$525 x$

+

$21 x^{3}$

-

$0$

+

$525 x$

Step 6.17

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

$3 x^{3}$

+

$6 x^{2}$

-

$21 x$

$x^{2}$

+

$0 x$

+

$25$

$3 x^{5}$

+

$6 x^{4}$

+

$54 x^{3}$

+

$108 x^{2}$

-

$525 x$

-

$1050$

-

$3 x^{5}$

-

$0$

-

$75 x^{3}$

+

$6 x^{4}$

-

$21 x^{3}$

+

$108 x^{2}$

-

$6 x^{4}$

-

$0$

-

$150 x^{2}$

-

$21 x^{3}$

-

$42 x^{2}$

-

$525 x$

+

$21 x^{3}$

-

$0$

+

$525 x$

-

$42 x^{2}$

+

$0$

Step 6.18

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

							$3 x^{3}$	+	$6 x^{2}$	-	$21 x$
	$x^{2}$	+	$0 x$	+	$25$		$3 x^{5}$	+	$6 x^{4}$	+	$54 x^{3}$	+	$108 x^{2}$	-	$525 x$	-	$1050$
						-	$3 x^{5}$	-	$0$	-	$75 x^{3}$
								+	$6 x^{4}$	-	$21 x^{3}$	+	$108 x^{2}$
								-	$6 x^{4}$	-	$0$	-	$150 x^{2}$
										-	$21 x^{3}$	-	$42 x^{2}$	-	$525 x$
										+	$21 x^{3}$	-	$0$	+	$525 x$
												-	$42 x^{2}$	+	$0$	-	$1050$

Step 6.19

Divide the highest order term in the dividend $- 42 x^{2}$ by the highest order term in divisor $x^{2}$ .

							$3 x^{3}$	+	$6 x^{2}$	-	$21 x$	-	$42$
	$x^{2}$	+	$0 x$	+	$25$		$3 x^{5}$	+	$6 x^{4}$	+	$54 x^{3}$	+	$108 x^{2}$	-	$525 x$	-	$1050$
						-	$3 x^{5}$	-	$0$	-	$75 x^{3}$
								+	$6 x^{4}$	-	$21 x^{3}$	+	$108 x^{2}$
								-	$6 x^{4}$	-	$0$	-	$150 x^{2}$
										-	$21 x^{3}$	-	$42 x^{2}$	-	$525 x$
										+	$21 x^{3}$	-	$0$	+	$525 x$
												-	$42 x^{2}$	+	$0$	-	$1050$

Step 6.20

Multiply the new quotient term by the divisor.

							$3 x^{3}$	+	$6 x^{2}$	-	$21 x$	-	$42$
	$x^{2}$	+	$0 x$	+	$25$		$3 x^{5}$	+	$6 x^{4}$	+	$54 x^{3}$	+	$108 x^{2}$	-	$525 x$	-	$1050$
						-	$3 x^{5}$	-	$0$	-	$75 x^{3}$
								+	$6 x^{4}$	-	$21 x^{3}$	+	$108 x^{2}$
								-	$6 x^{4}$	-	$0$	-	$150 x^{2}$
										-	$21 x^{3}$	-	$42 x^{2}$	-	$525 x$
										+	$21 x^{3}$	-	$0$	+	$525 x$
												-	$42 x^{2}$	+	$0$	-	$1050$
												-	$42 x^{2}$	+	$0$	-	$1050$

Step 6.21

The expression needs to be subtracted from the dividend, so change all the signs in $- 42 x^{2} + 0 - 1050$

							$3 x^{3}$	+	$6 x^{2}$	-	$21 x$	-	$42$
	$x^{2}$	+	$0 x$	+	$25$		$3 x^{5}$	+	$6 x^{4}$	+	$54 x^{3}$	+	$108 x^{2}$	-	$525 x$	-	$1050$
						-	$3 x^{5}$	-	$0$	-	$75 x^{3}$
								+	$6 x^{4}$	-	$21 x^{3}$	+	$108 x^{2}$
								-	$6 x^{4}$	-	$0$	-	$150 x^{2}$
										-	$21 x^{3}$	-	$42 x^{2}$	-	$525 x$
										+	$21 x^{3}$	-	$0$	+	$525 x$
												-	$42 x^{2}$	+	$0$	-	$1050$
												+	$42 x^{2}$	-	$0$	+	$1050$

Step 6.22

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

							$3 x^{3}$	+	$6 x^{2}$	-	$21 x$	-	$42$
	$x^{2}$	+	$0 x$	+	$25$		$3 x^{5}$	+	$6 x^{4}$	+	$54 x^{3}$	+	$108 x^{2}$	-	$525 x$	-	$1050$
						-	$3 x^{5}$	-	$0$	-	$75 x^{3}$
								+	$6 x^{4}$	-	$21 x^{3}$	+	$108 x^{2}$
								-	$6 x^{4}$	-	$0$	-	$150 x^{2}$
										-	$21 x^{3}$	-	$42 x^{2}$	-	$525 x$
										+	$21 x^{3}$	-	$0$	+	$525 x$
												-	$42 x^{2}$	+	$0$	-	$1050$
												+	$42 x^{2}$	-	$0$	+	$1050$
																	$0$

Step 6.23

Since the remander is $0$ , the final answer is the quotient.

$3 x^{3} + 6 x^{2} - 21 x - 42$

Step 6.24

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

$y = 3 x^{3} + 6 x^{2} - 21 x$

$y = 3 x^{3} + 6 x^{2} - 21 x$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = 3 x^{3} + 6 x^{2} - 21 x$

Step 8

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$