91Ó°ÊÓ

Simplify the rational expression by using long division or synthetic division. $$\frac{x^{4}+9 x^{3}-5 x^{2}-36 x+4}{x^{2}-4}$$

A company that produces calculators estimated that the profit \(P\) (in dollars) from selling a particular model of calculator was $$P=-152 x^{3}+7545 x^{2}-169,625, \quad 0 \leq x \leq 45$$ where \(x\) was the advertising expense (in tens of thousands of dollars). For this model of calculator, the advertising expense was \(\$ 400,000(x=40)\) and the profit was \(\$ 2,174,375.\) (a) Use a graphing utility to graph the profit function. (b) Use the graph from part (a) to estimate another amount the company could have spent on advertising that would have produced the same profit. (c) Use synthetic division to confirm the result of part (b) algebraically.

Sketching the Graph of a Polynomial Function Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the real zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. $$h(x)=\frac{1}{3} x^{3}(x-4)^{2}$$

Briefly explain what it means for a divisor to divide evenly into a dividend.

Briefly explain how to check polynomial division, and justify your reasoning. Give an example.

Assume that the function $$f(x)=a x^{2}+b x+c, \quad a \neq 0$$ has two real zeros. Prove that the \(x\) -coordinate of the vertex of the graph is the average of the zeros of \(f\) (Hint: Use the Quadratic Formula.)

Proof Prove that the complex conjugate of the product of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the product of their complex conjugates.

The revenues \(R\) (in millions of dollars) for a company from 2003 through 2010 can be modeled by \(R=6.212 t^{3}-132.87 t^{2}+863.2 t-1115,3 \leq t \leq 10\) where \(t\) represents the year, with \(t=3\) corresponding to 2003 (a) Use a graphing utility to approximate any relative extreme of the model over its domain. (b) Use the graphing utility to approximate the intervals on which the revenue for the company is increasing and decreasing over its domain. (c) Use the results of parts (a) and (b) to describe the company's revenue during this time period.

Find the rational zeros of the polynomial function. $$f(z)=z^{3}+\frac{11}{6} z^{2}-\frac{1}{2} z-\frac{1}{3}=\frac{1}{6}\left(6 z^{3}+11 z^{2}-3 z-2\right)$$

True or False? Determine whether the statement is true or false. Justify your answer. If the graph of a polynomial function falls to the right, then its leading coefficient is negative.