Chapter 3

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

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=2 x^{5}-x^{4}+2 x^{3}-2 x^{2}+4 x-4\); no real zero greater than 1

Divide. $$\frac{x^{3}-x^{2}+2 x-3}{x^{2}+3}$$

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 \(-1\). $$\begin{aligned} &1\\\ &\frac{1}{i^{-46}} \end{aligned}$$

Divide. $$\frac{8 x^{3}+10 x^{2}-12 x-15}{2 x^{2}-3}$$

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 \(-1\). $$\frac{-1}{-i^{12}}$$

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=x^{4}+x^{3}-x^{2}+3\); no real zero less than \(-2\)

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=x^{5}+2 x^{3}-2 x^{2}+5 x+5\); no real zero less than \(-1\)

Divide. $$\frac{3 x^{4}-2 x^{2}-5}{3 x^{2}-5}$$

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 \(-1\). $$\frac{-1}{-i^{15}}$$

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=3 x^{4}+2 x^{3}-4 x^{2}+x-1\); no real zero greater than 1

Divide. $$\frac{2 x^{4}-x^{3}+4 x^{2}+8 x+7}{2 x^{2}+3 x+2}$$

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

Show that \(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\) is a square root of \(i\).

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=3 x^{4}+2 x^{3}-4 x^{2}+x-1\); no real zero less than \(-2\)

Divide. $$\frac{3 x^{4}+2 x^{3}-x^{2}+4 x-3}{x^{2}+x-1}$$

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

Show that \(\frac{\sqrt{3}}{2}+\frac{1}{2} i\) is a cube root of \(i\).

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=x^{5}-3 x^{3}+x+2\); no real zero greater than 2

Divide. $$\left(x^{2}+\frac{1}{2} x-1\right) \div(2 x+1)$$

Find the conjugate of each number. $$5-3 i$$

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=x^{5}-3 x^{3}+x+2\); no real zero less than \(-3\)

Divide. $$\left(-x^{2}-1\right) \div(3 x-9)$$

Find the conjugate of each number. $$-3+i$$

Divide. $$\left(x^{3}-x^{2}+1\right) \div\left(2 x^{2}-1\right)$$

Find the conjugate of each number. $$-18 i$$

Divide. $$\left(-3 x^{3}+2 x^{2}+2 x\right) \div\left(6 x^{2}+2 x+1\right)$$

Find the conjugate of each number. $$\sqrt{7}$$

For each polynomial function, do the following in order. (a) Use Descartes' rule of signs to find the possible number of positive and negative real zeros. (b) Use the rational zeros theorem to determine the possible rational zeros of the function. (c) Find the rational zeros, if any. (d) Find all other real zeros, if any. (e) Find any other nonreal complex zeros, if any. (f) Find the \(x\) -intercepts of the graph, if any. (g) Find the \(y\) -intercept of the graph. (h) Use synthetic division to find \(P(4),\) and give the coordinates of the corresponding point on the graph. (i) Determine the end behavior of the graph. (i) Sketch the graph. (You may wish to support your answer with a calculator graph.) $$P(x)=-2 x^{4}-x^{3}+x+2$$

Find the conjugate of each number. $$-\sqrt{8}$$

For each polynomial function, do the following in order. (a) Use Descartes' rule of signs to find the possible number of positive and negative real zeros. (b) Use the rational zeros theorem to determine the possible rational zeros of the function. (c) Find the rational zeros, if any. (d) Find all other real zeros, if any. (e) Find any other nonreal complex zeros, if any. (f) Find the \(x\) -intercepts of the graph, if any. (g) Find the \(y\) -intercept of the graph. (h) Use synthetic division to find \(P(4),\) and give the coordinates of the corresponding point on the graph. (i) Determine the end behavior of the graph. (i) Sketch the graph. (You may wish to support your answer with a calculator graph.) $$P(x)=4 x^{5}+8 x^{4}+9 x^{3}+27 x^{2}+27 x$$

Find the conjugate of each number. $$8 i$$

For each polynomial function, do the following in order. (a) Use Descartes' rule of signs to find the possible number of positive and negative real zeros. (b) Use the rational zeros theorem to determine the possible rational zeros of the function. (c) Find the rational zeros, if any. (d) Find all other real zeros, if any. (e) Find any other nonreal complex zeros, if any. (f) Find the \(x\) -intercepts of the graph, if any. (g) Find the \(y\) -intercept of the graph. (h) Use synthetic division to find \(P(4),\) and give the coordinates of the corresponding point on the graph. (i) Determine the end behavior of the graph. (i) Sketch the graph. (You may wish to support your answer with a calculator graph.) $$P(x)=3 x^{4}-14 x^{2}-5$$

Divide as indicated. Write each quotient in standard form. $$\frac{3}{-1}$$

For each polynomial function, do the following in order. (a) Use Descartes' rule of signs to find the possible number of positive and negative real zeros. (b) Use the rational zeros theorem to determine the possible rational zeros of the function. (c) Find the rational zeros, if any. (d) Find all other real zeros, if any. (e) Find any other nonreal complex zeros, if any. (f) Find the \(x\) -intercepts of the graph, if any. (g) Find the \(y\) -intercept of the graph. (h) Use synthetic division to find \(P(4),\) and give the coordinates of the corresponding point on the graph. (i) Determine the end behavior of the graph. (i) Sketch the graph. (You may wish to support your answer with a calculator graph.) $$P(x)=-x^{5}-x^{4}+10 x^{3}+10 x^{2}-9 x-9$$

Divide as indicated. Write each quotient in standard form. $$\frac{-7}{3 i}$$

For each polynomial function, do the following in order. (a) Use Descartes' rule of signs to find the possible number of positive and negative real zeros. (b) Use the rational zeros theorem to determine the possible rational zeros of the function. (c) Find the rational zeros, if any. (d) Find all other real zeros, if any. (e) Find any other nonreal complex zeros, if any. (f) Find the \(x\) -intercepts of the graph, if any. (g) Find the \(y\) -intercept of the graph. (h) Use synthetic division to find \(P(4),\) and give the coordinates of the corresponding point on the graph. (i) Determine the end behavior of the graph. (i) Sketch the graph. (You may wish to support your answer with a calculator graph.) $$P(x)=-3 x^{4}+22 x^{3}-55 x^{2}+52 x-12$$

Divide as indicated. Write each quotient in standard form. $$\frac{-10}{i}$$

Divide as indicated. Write each quotient in standard form. $$\frac{-19-9 i}{i}$$

Divide as indicated. Write each quotient in standard form. $$\frac{1-3 i}{1+i}$$

Divide as indicated. Write each quotient in standard form. $$\frac{-12-5 i}{3-2 i}$$

Divide as indicated. Write each quotient in standard form. $$\frac{-3+4 i}{2-i}$$

Divide as indicated. Write each quotient in standard form. $$\frac{-6+8 i}{1-i}$$

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. $$(a) x^{2}+4 x+3 \geq 0$$ $$(b) x^{2}+4 x+3<0$$

Divide as indicated. Write each quotient in standard form. $$\frac{4-3 i}{4+3 i}$$

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. $$(a) x^{2}+6 x+8<0$$ $$(b)x^{2}+6 x+8 \geq 0$$

Divide as indicated. Write each quotient in standard form. $$\frac{2-i}{2+i}$$