Find the Asymptotes (2x^2-5x+2)/(x-3)

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

Step 1

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

$x = 3$

Step 2

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 3

Find $n$ and $m$ .

$n = 2$

$m = 1$

Step 4

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

No Horizontal Asymptotes

Step 5

Find the oblique asymptote using polynomial division.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

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

Rewrite $- 5$ as $- 1$ plus $- 4$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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

Expand $(2 x - 1) (x - 2)$ .

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Move $x$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $- 2$ .

$\frac{2 x^{2} - 4 x - 1 x - 1 \cdot - 2}{x - 3}$

Multiply $- 1$ by $- 2$ .

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

Subtract $1 x$ from $- 4 x$ .

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

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

$x$

$3$

$2 x^{2}$

$5 x$

$2$

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

					$2 x$
	$x$	-	$3$		$2 x^{2}$	-	$5 x$	+	$2$

Multiply the new quotient term by the divisor.

				$2 x$
$x$	-	$3$		$2 x^{2}$	-	$5 x$	+	$2$
			+	$2 x^{2}$	-	$6 x$

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

				$2 x$
$x$	-	$3$		$2 x^{2}$	-	$5 x$	+	$2$
			-	$2 x^{2}$	+	$6 x$

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

				$2 x$
$x$	-	$3$		$2 x^{2}$	-	$5 x$	+	$2$
			-	$2 x^{2}$	+	$6 x$
					+	$x$

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

				$2 x$
$x$	-	$3$		$2 x^{2}$	-	$5 x$	+	$2$
			-	$2 x^{2}$	+	$6 x$
					+	$x$	+	$2$

Divide the highest order term in the dividend $x$ by the highest order term in divisor $x$ .

				$2 x$	+	$1$
$x$	-	$3$		$2 x^{2}$	-	$5 x$	+	$2$
			-	$2 x^{2}$	+	$6 x$
					+	$x$	+	$2$

Multiply the new quotient term by the divisor.

				$2 x$	+	$1$
$x$	-	$3$		$2 x^{2}$	-	$5 x$	+	$2$
			-	$2 x^{2}$	+	$6 x$
					+	$x$	+	$2$
					+	$x$	-	$3$

The expression needs to be subtracted from the dividend, so change all the signs in $x - 3$

				$2 x$	+	$1$
$x$	-	$3$		$2 x^{2}$	-	$5 x$	+	$2$
			-	$2 x^{2}$	+	$6 x$
					+	$x$	+	$2$
					-	$x$	+	$3$

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

				$2 x$	+	$1$
$x$	-	$3$		$2 x^{2}$	-	$5 x$	+	$2$
			-	$2 x^{2}$	+	$6 x$
					+	$x$	+	$2$
					-	$x$	+	$3$
							+	$5$

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

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

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

$y = 2 x + 1$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 3$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 2 x + 1$

Step 7

Find the Asymptotes (2x^2-5x+2)/(x-3)

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