91Ó°ÊÓ

Use algebra to determine the location of the vertical asymptotes and holes in the graph of the function. $$g(x)=\frac{x^{3}+5 x^{2}+8 x+4}{x^{3}+4 x^{2}+5 x+2}$$

Without graphing, determine the vertex of the given parabola and state whether it opens upward or downward. $$y=3 x^{2}+6 x+1$$

Find all the roots of \(f(x)\) in the complex number system; then write \(f(x)\) as a product of linear factors. $$f(x)=x^{3}+125$$

State the quotient and remainder when the first polynomial is divided by the second. Check your division by calculating (Divisor)(Quotient) + Remainder. $$x^{5}-1 ; \quad x-1$$

In Exercises \(16-22,\) factor the polynomial as a product of lin. ear factors and a factor \(g(x)\) such that \(g(x)\) is either a constant or a polynomial that has no rational roots. $$12 x^{3}-10 x^{2}+6 x-2$$

In Exercises \(1-54,\) perform the indicated operation and write the result in the form \(a+b i\). $$i^{26}$$

Determine whether the first polynomial is a factor of the second. $$x^{2}+5 x-1 ; \quad x^{3}+2 x^{2}-5 x-6$$

In Exercises \(16-22,\) factor the polynomial as a product of lin. ear factors and a factor \(g(x)\) such that \(g(x)\) is either a constant or a polynomial that has no rational roots. $$x^{6}-4 x^{5}+3 x^{4}-12 x^{3}$$

Find the horizontal asymptote, if any, of the graph of the given function. If there is a horizontal asymptote, find a viewing window in which the ends of the graph are within .1 of this asymptote. $$f(x)=\frac{x+1}{x^{6}+20}$$

In Exercises \(1-54,\) perform the indicated operation and write the result in the form \(a+b i\). $$i^{33}$$