Question

question #5
jeannie determined the equation of the slant asymptote to be y = 2x - 3 for the given rational function f(x)=\frac{2x^{2}-x + 4}{x + 1}. review jeannies work and determine if she is correct. if not, state the step and error she first made.
step 1: determine if has a slant asymptote. the degree of the numerator is greater than the degree of the denominator.
step 2: set up for synthetic division. -1 2 -1 4
step 3: perform synthetic division. -1 2 -1 4 -2 3 2 -3 7
step 4: write the equation of the asymptote. slant asymptote: y = 2x - 1
jeannie is incorrect. she made an error in step 2 when putting the divisor on the outside of the dividend.
jeannie is correct, there are no errors in her work.
jeannie is incorrect. she made an error in step 3 when performing synthetic division.
jeannie is incorrect. she made an error in step 4 when she wrote the equation of the slant asymptote.

Explanation:

Step1: Recall slant - asymptote condition

A rational function $F(x)=\frac{f(x)}{g(x)}$ has a slant asymptote when the degree of the numerator $f(x)$ is one more than the degree of the denominator $g(x)$. Here, for $F(x)=\frac{2x^{2}-x + 4}{x + 1}$, the degree of the numerator is 2 and the degree of the denominator is 1, so it has a slant asymptote.

Step2: Perform synthetic division

The divisor for synthetic division of $\frac{2x^{2}-x + 4}{x + 1}$ is $- 1$ (since $x+1=0$ gives $x=-1$). The coefficients of the numerator are $2,-1,4$.
Set up synthetic division:

-1 |  2  -1  4
    |     -2  3
    |__________
      2  -3  7

The quotient is $2x-3$ and the remainder is 7.

Step3: Write the equation of the slant asymptote

The equation of the slant asymptote is given by the quotient of the polynomial long - division (or synthetic division in this case). The slant asymptote of $F(x)=\frac{2x^{2}-x + 4}{x + 1}$ is $y = 2x-3$.

Answer:

Jeannie is correct, there are no errors in her work.