Slant Asymptotes

​Slant Asymptotes

Parker - Precalculus

Multiple Choice

Divide according to the notes from yesterday:

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

• $$2x-8+\frac{13}{x+1}$$

• $$2x-4+\frac{1}{x+1}$$

• $$2x-8-\frac{3}{x+1}$$

• $$2x-4+\frac{9}{x+1}$$

​Recall the rules for horizontal asymptotes:

n < m: Horizontal Asymptote at y=0

n = m: Horizontal Asymptote at the ratio of leading coefficients

n > m: Slant Asymptote

​You may notice that this looks like the equation of a line in y = mx + b form...

...if you did, you’d be right! That’s why the asymptote line is slanted!
Since the asymptote that is produced is a linear equation, it can be graphed as one:

Multiple Choice

Practice one more example of long division to find the slant asymptote of the function:

$$f\left(x\right)=\frac{x^2+5x+6}{x-3}$$

• $$x+8$$

• $$x+2$$

• $$x+5$$

• $$x+4$$

Need to find slant asymptotes?
Polynomial division got you confused?
There's GOT to be a better way!

There is!

​Now introducing: Synthetic Division

Synthetic division replaces all of that old, run-down distribution in each step and replaces each term with its coefficient!

Watch out though! This only works when we are dividing our polynomial by a linear expression, i.e. by a first-degree binomial.

Let's look at an example:

​How does it work?

With long division, you divided the leading term of the divisor into each term and remainder of the dividend; with synthetic division, you take the constant term of the divisor and multiply it by each coefficient of the dividend. Look how much shorter that is!

Let's take this step-by-step with an example problem:

Notice that the divisor has to be in (x - a) form. That means that, like in this example, a term in the form (x + a) must be turned into subtraction of a negative.

When we write our equation, we use something like an upside-down long division sign. Notice that there is space underneath our coefficients! We need that space!

Multiply each number in the solution row by the divisor, then add to the next coefficient.

Multiply each number in the solution row by the divisor, then add to the next coefficient.

Multiply each number in the solution row by the divisor, then add to the next coefficient.

Multiply each number in the solution row by the divisor, then add to the next coefficient.

Multiply each number in the solution row by the divisor, then add to the next coefficient.

Finally, identify and write the remainder as a fraction, with the divisor in the denominator.

Now that we have a quotient of the two polynomials, let's remember our goal here: the slant asymptote.

Notice that our quotient is one degree less (linear) than our original dividend (quadratic). This linear expression is our slant asymptote:

y= x + 3

Try a few examples. Pay attention to whether or not the instructions ask for the full quotient of the equation or just the slant asymptote.

Multiple Choice

Find the quotient of the rational expression:

$$\frac{3x^2+5x+10}{x-5}=$$

• $$3x+20+\frac{110}{x-5}$$

• $$3x+20$$

• $$3x-20-\frac{110}{x-5}$$

• $$3x-20$$

Multiple Choice

Find the slant asymptote of the function:

$$f\left(x\right)=\frac{3x^2+5x+10}{x-5}$$

• $$3x+20+\frac{110}{x-5}$$

• $$3x+20$$

• $$3x-20-\frac{110}{x-5}$$

• $$3x-20$$

Multiple Choice

Find the quotient of the rational expression:

$$\frac{5x^2+18x+45}{x-3}=$$

• $$5x+33+\frac{144}{x-3}$$

• $$5x+33$$

• $$5x+3+\frac{36}{x-3}$$

• $$5x+3$$

Multiple Choice

Find the slant asymptote of the function:

$$\frac{5x^2+18x+45}{x-3}=$$

• $$5x+33+\frac{144}{x-3}$$

• $$5x+33$$

• $$5x+3+\frac{36}{x-3}$$

• $$5x+3$$