Write an equation of the slant or nonlinear asymptot \[ f(x)=\frac{-4 x^{3}+8 x^{2}-2 x+5}{x-2} \] The (Choose one) \( \nabla \) asymptote is \( y=\square \). slant nonlinear

To find the slant asymptote for the function \( f(x) = \frac{-4x^{3} + 8x^{2} - 2x + 5}{x - 2} \), we can perform polynomial long division. Dividing \( -4x^{3} + 8x^{2} - 2x + 5 \) by \( x - 2 \) gives us a quotient of \( -4x^{2} - 0x - 8 \). The slant asymptote is the result of the quotient, which simplifies to \( y = -4x^{2} - 8 \). So, the slant asymptote is \( y = -4x^{2} - 8 \). Now, keep in mind that slant asymptotes occur when the degree of the numerator is exactly one higher than the degree of the denominator. Thus, a clear understanding of polynomial long division can greatly help identify such asymptotes in other functions as well! Polynomials are fun because there’s a whole world of their behavior you can explore, including roots, and even turning points!