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Chapter 2: Problem 54
Write the quotient in standard form. $$\frac{13}{1-i}$$
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Find the rational zeros of the polynomial function. $$P(x)=x^{4}-\frac{25}{4} x^{2}+9=\frac{1}{4}\left(4 x^{4}-25 x^{2}+36\right)$$
Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{4}+10 x^{2}+9$$
Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-2 x^{3}-3 x^{2}+12 x-18\) (Hint: One factor is \(\left.x^{2}-6 .\right)\)
cost The ordering and transportation cost \(C\) (in thousands of dollars) for machine parts is \(C=100\left(\frac{200}{x^{2}}+\frac{x}{x+30}\right), \quad x \geq 1\) where \(x\) is the order size (in hundreds). In calculus, it can be shown that the cost is a minimum when \(3 x^{3}-40 x^{2}-2400 x-36,000=0\) Use a calculator to approximate the optimal order size to the nearest hundred units.
(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$2 x^{2}+b x+5=0$$
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