Question

fill in the blank so that the resulting statement is true. based on the synthetic division shown below, the equation of the slant asymptote of $f(x)=\frac{5x^2 - 2x + 4}{x - 3}$ is ______ \\(\

$$\begin{array}{r|rrr}3 & 5 & -2 & 4 \\\\ & & 15 & 39 \\\\ \\hline & 5 & 13 & 43\\end{array}$$

\\) based on the synthetic division shown, the equation of the slant asymptote of $f(x)=\frac{5x^2 - 2x + 4}{x - 3}$ is \\(\square\\). (type an equation.)

Explanation:

Step1: Recall Slant Asymptote Rule

For a rational function \( f(x)=\frac{N(x)}{D(x)} \) where degree of \( N(x) \) is 1 more than degree of \( D(x) \), slant asymptote is found by dividing \( N(x) \) by \( D(x) \); the quotient (ignoring remainder) gives the slant asymptote.

Step2: Interpret Synthetic Division

Synthetic division of \( 5x^2 - 2x + 4 \) by \( x - 3 \) gives quotient \( 5x + 13 \) (from coefficients 5, 13) and remainder 43. So \( f(x)=5x + 13+\frac{43}{x - 3} \).

Step3: Determine Slant Asymptote

As \( x\to\pm\infty \), \( \frac{43}{x - 3}\to0 \), so the slant asymptote is \( y = 5x + 13 \).

Answer:

\( y = 5x + 13 \)