Pre-Algebra Examples

$\frac{y^{2} - 2 y + 1}{2 y - 3}$

Step 1

Find where the expression $\frac{{(y - 1)}^{2}}{2 y - 3}$ is undefined.

$y = \frac{3}{2}$

Step 2

$x = \frac{{(y - 1)}^{2}}{2 y - 3}$ is an equation of a line, which means there are no horizontal asymptotes.

No Horizontal Asymptotes

Step 3

Find the oblique asymptote using polynomial division.

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Step 3.1

Factor using the perfect square rule.

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Step 3.1.1

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

$\frac{y^{2} - 2 y + 1^{2}}{2 y - 3}$

Step 3.1.2

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

$2 y = 2 \cdot y \cdot 1$

Step 3.1.3

Rewrite the polynomial.

$\frac{y^{2} - 2 \cdot y \cdot 1 + 1^{2}}{2 y - 3}$

Step 3.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 1$ .

$\frac{{(y - 1)}^{2}}{2 y - 3}$

Step 3.2

Expand ${(y - 1)}^{2}$ .

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Step 3.2.1

Rewrite ${(y - 1)}^{2}$ as $(y - 1) (y - 1)$ .

$\frac{(y - 1) (y - 1)}{2 y - 3}$

Step 3.2.2

Apply the distributive property.

$\frac{y (y - 1) - 1 (y - 1)}{2 y - 3}$

Step 3.2.3

Apply the distributive property.

$\frac{y \cdot y + y \cdot - 1 - 1 (y - 1)}{2 y - 3}$

Step 3.2.4

Apply the distributive property.

$\frac{y \cdot y + y \cdot - 1 - 1 y - 1 \cdot - 1}{2 y - 3}$

Step 3.2.5

Reorder $y$ and $- 1$ .

$\frac{y \cdot y - 1 y - 1 y - 1 \cdot - 1}{2 y - 3}$

Step 3.2.6

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

$\frac{y \cdot y - 1 y - 1 y - 1 \cdot - 1}{2 y - 3}$

Step 3.2.7

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

$\frac{y \cdot y - 1 y - 1 y - 1 \cdot - 1}{2 y - 3}$

Step 3.2.8

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

$\frac{y^{1 + 1} - 1 y - 1 y - 1 \cdot - 1}{2 y - 3}$

Step 3.2.9

Add $1$ and $1$ .

$\frac{y^{2} - 1 y - 1 y - 1 \cdot - 1}{2 y - 3}$

Step 3.2.10

Multiply $- 1$ by $- 1$ .

$\frac{y^{2} - 1 y - 1 y + 1}{2 y - 3}$

Step 3.2.11

Subtract $1 y$ from $- 1 y$ .

$\frac{y^{2} - 2 y + 1}{2 y - 3}$

Step 3.3

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

$2 y$

$3$

$y^{2}$

$2 y$

$1$

Step 3.4

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

					$\frac{y}{2}$
	$2 y$	-	$3$		$y^{2}$	-	$2 y$	+	$1$

Step 3.5

Multiply the new quotient term by the divisor.

				$\frac{y}{2}$
$2 y$	-	$3$		$y^{2}$	-	$2 y$	+	$1$
			+	$y^{2}$	-	$\frac{3 y}{2}$

Step 3.6

The expression needs to be subtracted from the dividend, so change all the signs in $y^{2} - \frac{3 y}{2}$

				$\frac{y}{2}$
$2 y$	-	$3$		$y^{2}$	-	$2 y$	+	$1$
			-	$y^{2}$	+	$\frac{3 y}{2}$

Step 3.7

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

				$\frac{y}{2}$
$2 y$	-	$3$		$y^{2}$	-	$2 y$	+	$1$
			-	$y^{2}$	+	$\frac{3 y}{2}$
					-	$\frac{y}{2}$

Step 3.8

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

				$\frac{y}{2}$
$2 y$	-	$3$		$y^{2}$	-	$2 y$	+	$1$
			-	$y^{2}$	+	$\frac{3 y}{2}$
					-	$\frac{y}{2}$	+	$1$

Step 3.9

Divide the highest order term in the dividend $- \frac{y}{2}$ by the highest order term in divisor $2 y$ .

				$\frac{y}{2}$	-	$\frac{1}{4}$
$2 y$	-	$3$		$y^{2}$	-	$2 y$	+	$1$
			-	$y^{2}$	+	$\frac{3 y}{2}$
					-	$\frac{y}{2}$	+	$1$

Step 3.10

Multiply the new quotient term by the divisor.

				$\frac{y}{2}$	-	$\frac{1}{4}$
$2 y$	-	$3$		$y^{2}$	-	$2 y$	+	$1$
			-	$y^{2}$	+	$\frac{3 y}{2}$
					-	$\frac{y}{2}$	+	$1$
					-	$\frac{y}{2}$	+	$\frac{3}{4}$

Step 3.11

The expression needs to be subtracted from the dividend, so change all the signs in $- \frac{y}{2} + \frac{3}{4}$

				$\frac{y}{2}$	-	$\frac{1}{4}$
$2 y$	-	$3$		$y^{2}$	-	$2 y$	+	$1$
			-	$y^{2}$	+	$\frac{3 y}{2}$
					-	$\frac{y}{2}$	+	$1$
					+	$\frac{y}{2}$	-	$\frac{3}{4}$

Step 3.12

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

				$\frac{y}{2}$	-	$\frac{1}{4}$
$2 y$	-	$3$		$y^{2}$	-	$2 y$	+	$1$
			-	$y^{2}$	+	$\frac{3 y}{2}$
					-	$\frac{y}{2}$	+	$1$
					+	$\frac{y}{2}$	-	$\frac{3}{4}$
							+	$\frac{1}{4}$

Step 3.13

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

$\frac{y}{2} - \frac{1}{4} + \frac{1}{4 (2 y - 3)}$

Step 3.14

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

$y = \frac{y}{2} - \frac{1}{4}$

Step 4

This is the set of all asymptotes.

Vertical Asymptotes: $y = \frac{3}{2}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{y}{2} - \frac{1}{4}$

Step 5