Write an equation of the slant or nonlinear asymptote. $$ n(x)=\frac{4 x^{3}+5 x^{2}-2 x+2}{x^{2}+4} $$ The (Choose one) $$ \nabla $$ asymptote is $$ y=\square $$ slant nonlinear

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UpStudy AI · Accepted Answer

Perform polynomial long division of 
$$
\frac{4x^3+5x^2-2x+2}{x^2+4}
$$
by dividing the numerator by the denominator.

**Step 1.** Divide the leading term of the numerator by the leading term of the denominator:
$$
\frac{4x^3}{x^2} = 4x.
$$
Multiply the divisor by $$4x$$:
$$
4x(x^2+4)=4x^3+16x.
$$
Subtract this from the original numerator:
$$
(4x^3+5x^2-2x+2) - (4x^3+16x)=5x^2-18x+2.
$$

**Step 2.** Divide the new leading term by the leading term of the divisor:
$$
\frac{5x^2}{x^2}=5.
$$
Multiply the divisor by $$5$$:
$$
5(x^2+4)=5x^2+20.
$$
Subtract from the current remainder:
$$
(5x^2-18x+2) - (5x^2+20)=-18x-18.
$$

The division gives:
$$
n(x)=4x+5+\frac{-18x-18}{x^2+4}.
$$

Since the fraction $$\frac{-18x-18}{x^2+4}$$ tends to $$0$$ as $$x\to\pm\infty$$, the slant asymptote is the quotient:
$$
y=4x+5.
$$
The slant asymptote is $$ y = 4x + 5 $$.

Write an equation of the slant or nonlinear asymptote. \[ n(x)=\frac{4 x^{3}+5 x^{2}-2 x+2}{x^{2}+4} \] The (Choose one) \( \nabla \) asymptote is \( y=\square \) slant nonlinear

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