91Ó°ÊÓ

Find a polynomial f(x) with real coefficients that satisfies the given conditions. Some of these problems have many correct answers. Degree \(2 ;\) roots \(1+2 i\) and \(1-2 i\)

In Exercises \(1-54,\) perform the indicated operation and write the result in the form \(a+b i\). $$\frac{i}{3+i}-\frac{3+i}{4+i}$$

Find the remainder when \(f(x)\) is divided by \(g(x),\) without using division. $$f(x)=x^{3}+8 x^{2}-29 x+44 ; \quad g(x)=x+11$$

Solve the inequality. Find exact solutions when possible and approximate ones otherwise. $$x^{5}+5 x^{4} > 4 x^{3}-3 x^{2}+2$$

Find a polynomial f(x) with real coefficients that satisfies the given conditions. Some of these problems have many correct answers. Degree \(4 ;\) roots \(3 i\) and \(-3 i,\) each of multiplicity 2

Analyze the function algebraically. List its vertical asymptotes, holes, y-intercept, and horizontal asymptote, if any. Then sketch a complete graph of the function. $$f(x)=\frac{(x-1)(x-2)(x-3)}{(4 x+1)}$$

In Exercises \(1-54,\) perform the indicated operation and write the result in the form \(a+b i\). $$6+\frac{2 i}{3+i}$$

$$\text { In Exercises } 29-40, \text { find all real roots of the polynomial.}$$ $$\begin{array}{l} x^{10}-10 x^{9}+45 x^{8}-120 x^{7}+210 x^{6}-252 x^{5}+210 x^{4}- \\ 120 x^{3}+45 x^{2}-10 x+1 \end{array}$$

Find the rule of the quadratic function whose graph passes through the given points (one of which is the vertex. $$(3.1,4.1),(6.1,13.1),(-.9,20.1)$$

Find a polynomial f(x) with real coefficients that satisfies the given conditions. Some of these problems have many correct answers. Degree \(4 ;\) only roots are \(4,3+i,\) and \(3-i\)