Chapter 4

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. $$P(x)=x^{5}+3 x^{4}-x^{3}+2 x+3$$

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

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. $$P(x)=2 x^{5}-x^{4}+x^{3}-x^{2}+x+5$$

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=x^{4}-x^{3}+3 x^{2}-8 x+8\) no real zero greater than 2

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

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

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 \(-2\)

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

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

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

Divide. $$\frac{3 x^{4}+2 x^{3}-x^{2}+4 x-8}{x^{2}+x-1}$$

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

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

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

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