91Ó°ÊÓ

State the quotient and remainder when the first polynomial is divided by the second. Check your division by calculating (Divisor)(Quotient) + Remainder. $$2 x^{5}+5 x^{4}+x^{3}-7 x^{2}-13 x+12 ; \quad 2 x^{3}+x^{2}-7 x+4$$

In Exercises \(1-54,\) perform the indicated operation and write the result in the form \(a+b i\). $$(\sqrt{3}+i)(\sqrt{3}-i)$$

Directions: When asked to find the roots of a polynomial, find exact roots whenever possible and approximate the other roots. In Exercises \(1-15,\) find all the rational mots of the polynomial. $$\begin{array}{l} x^{10}-10 x^{9}+45 x^{8}-120 x^{7}+210 x^{6}-252 x^{5}+210 x^{4}- \\ 120 x^{3}+45 x^{2}-10 x+1 \end{array}$$

Use algebra to determine the location of the vertical asymptotes and holes in the graph of the function. $$f(x)=\frac{x^{3}+6 x^{2}+11 x+6}{x^{3}-x}$$

Find all the roots of \(f(x)\) in the complex number system; then write \(f(x)\) as a product of linear factors. $$f(x)=3 x^{2}+18 x+27$$

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. $$x^{15}-x-1$$

Without graphing, determine the vertex of the given parabola and state whether it opens upward or downward. $$f(x)=x^{2}-5 x-7$$

Find all the roots of \(f(x)\) in the complex number system; then write \(f(x)\) as a product of linear factors. $$f(x)=6 x^{2}-9 x-60$$

State the quotient and remainder when the first polynomial is divided by the second. Check your division by calculating (Divisor)(Quotient) + Remainder. $$3 x^{4}-3 x^{3}-11 x^{2}+6 x-1 ; \quad x^{3}+x^{2}-2$$

In Exercises \(1-54,\) perform the indicated operation and write the result in the form \(a+b i\). $$\left(\frac{1}{2}-i\right)\left(\frac{1}{4}+2 i\right)$$