91Ó°ÊÓ

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$\frac{1}{i^{9}}$$

Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros for \(P(x) .\) Then, use a graph to determine the actual numbers of positive and negative real zeros. $$P(x)=x^{5}+3 x^{4}-x^{3}+2 x+3$$

Concept Check For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverse.) $$2-\sqrt{5}, 2+\sqrt{5}$$

Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros for \(P(x) .\) Then, use a graph to determine the actual numbers of positive and negative real zeros. $$P(x)=2 x^{5}-x^{4}+x^{3}-x^{2}+x+5$$

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$\frac{1}{i^{12}}$$

Divide. $$\frac{20 x^{4}+6 x^{3}-2 x^{2}+15 x-2}{5 x-1}$$

Concept Check For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverse.) $$3 i,-3 i$$

Divide. $$\frac{5 x^{4}-2 x^{2}+6}{x^{2}+2}$$

Sketch a graph of \(f(x)=a x^{2}+b x+c\) that satisfies each set of conditions. $$a<0, b^{2}-4 a c=0$$

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$\frac{1}{i^{-51}}$$