Chapter 4

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=-3 x^{3}-x-5 ; \quad x+1$$

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. 5 (multiplicity 2 ) and \(-2 i\)

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&-4 x^{3}-x^{2}+4 x=0\\\&[-4,4] \text { by }[-10,10]\end{aligned}$$

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=2 x^{3}-5 x^{2}-x+6 \\ &=(x+1)(2 x-3)(x-2) \end{aligned}$$

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=x^{5}-1 ; x-1$$

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&3 x^{3}+3 x^{2}+3 x=0\\\&[-5,5] \text { by }[-5,5]\end{aligned}$$

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=2 x^{3}+9 x^{2}-6 x-40 \\ &=(x-2)(2 x+5)(x+4) \end{aligned}$$

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=x^{7}+1 ; \quad x+1$$

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&2 x^{3}+2 x^{2}+12 x=0\\\&[-10,10] \text { by }[-20,20]\end{aligned}$$

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=x^{4}-18 x^{2}+81 \\ &=(x-3)^{2}(x+3)^{2} \end{aligned}$$

Use synthetic substitution to find \(P(k).\) $$k=3 ; \quad P(x)=x^{2}-4 x+3$$

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&x^{4}+17 x^{2}+16=0\\\&[-4,4] \text { by }[-10,40]\end{aligned}$$

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=x^{4}-8 x^{2}+16 \\ &=(x+2)^{2}(x-2)^{2} \end{aligned}$$

Use synthetic substitution to find \(P(k).\) $$k=-2 ; \quad P(x)=x^{2}+5 x+6$$

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&36 x^{4}+85 x^{2}+9=0\\\&[-4,4] \text { by }[-10,40]\end{aligned}$$

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=2 x^{4}+x^{3}-6 x^{2}-7 x-2 \\ &=(2 x+1)(x-2)(x+1)^{2} \end{aligned}$$

Use synthetic substitution to find \(P(k).\) $$k=-2 ; \quad P(x)=5 x^{3}+2 x^{2}-x+5$$

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&x^{6}+19 x^{3}-216=0\\\&[-4,4] \text { by }[-350,200]\end{aligned}$$

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=3 x^{4}-7 x^{3}-6 x^{2}+12 x+8 \\ &=(3 x+2)(x+1)(x-2)^{2} \end{aligned}$$

Use synthetic substitution to find \(P(k).\) $$k=2 ; \quad P(x)=2 x^{3}-3 x^{2}-5 x+4$$

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&8 x^{6}+7 x^{3}-1=0\\\&[-4,4] \text { by }[-5,100]\end{aligned}$$

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=x^{4}+3 x^{3}-3 x^{2}-11 x-6 \\ &=(x+3)(x+1)^{2}(x-2) \end{aligned}$$

Use synthetic substitution to find \(P(k).\) $$k=2 ; \quad P(x)=x^{2}-5 x+1$$

Answer true or false to each statement. Then support your answer by graphing. The function \(f(x)=x^{3}+3 x^{2}+3 x+1\) must have at least one real zero.

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=-2 x^{5}+5 x^{4}+34 x^{3}-30 x^{2}-84 x+45 \\ &=(x+3)(2 x-1)(x-5)\left(3-x^{2}\right) \end{aligned}$$

Use synthetic substitution to find \(P(k).\) $$k=3 ; \quad P(x)=x^{2}-x+3$$

Graph each polynomial function by hand, as shown in the previous section. Then solve each equation or inequality. In pan (a), state if the multiplicity of a solution is greater than one. \(P(x)=x^{3}+4 x^{2}-11 x-30\) \(=(x-3)(x+2)(x+5)\) (a) \(P(x)=0 \quad\) (b) \(P(x)<0 \quad\) (c) \(P(x)>0\)

Answer true or false to each statement. Then support your answer by graphing. If a polynomial function of even degree has a negative leading coefficient and a positive \(y\) -value for its \(y\) -intercept, it must have at least two real zeros.

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=2 x^{5}-10 x^{4}+x^{3}-5 x^{2}-x+5 \\ &=(x-5)\left(x^{2}+1\right)\left(2 x^{2}-1\right) \end{aligned}$$

Use synthetic substitution to find \(P(k).\) $$k=0.5 ; \quad P(x)=x^{3}-x+4$$

Answer true or false to each statement. Then support your answer by graphing. The function \(f(x)=3 x^{4}+5\) has no real zeros.

Sketch by hand the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned} P(x) &=3 x^{4}-4 x^{3}-22 x^{2}+15 x+18 \\ &=(3 x+2)(x-3)\left(x^{2}+x-3\right) \end{aligned}$$

Use synthetic substitution to find \(P(k).\) $$k=1.5 ; \quad P(x)=x^{3}+x-3$$

Answer true or false to each statement. Then support your answer by graphing. The function \(f(x)=-3 x^{4}+5\) has two real zeros.

Use synthetic substitution to find \(P(k).\) $$k=\sqrt{2} ; \quad P(x)=x^{4}-x^{2}-3$$

Graph each polynomial function by hand, as shown in the previous section. Then solve each equation or inequality. In pan (a), state if the multiplicity of a solution is greater than one. \(P(x)=-x^{4}-4 x^{3}+3 x^{2}+18 x\) \(=-x(x-2)(x+3)^{2}\) (a) \(P(x)=0\) (b) \(P(x) \geq 0\) (c) \(P(x) \leq 0\)

Answer true or false to each statement. Then support your answer by graphing. The graph of \(f(x)=x^{3}-3 x^{2}+3 x-1=(x-1)^{3}\) has exactly one \(x\) -intercept.

Use synthetic substitution to find \(P(k).\) $$k=\sqrt{3} ; \quad P(x)=x^{4}+2 x^{2}-10$$

Answer true or false to each statement. Then support your answer by graphing. A fifth-degree polynomial function cannot have a single real zero.

Use synthetic substitution to find \(P(k).\) $$k=\sqrt[3]{4} ; \quad P(x)=-x^{3}+x+4$$

Solve each equation and inequality. (a) \(3\left(x^{2}+4\right)+2 x(3 x-12)=0\) (b) \(3\left(x^{2}+4\right)+2 x(3 x-12)<0\)

Use synthetic substitution to find \(P(k).\) $$k=\sqrt[5]{3} ; \quad P(x)=-x^{5}+2 x+3$$

Solve each equation and inequality. (a) \(\left(x^{2}+3 x-1\right)+(2 x+3)(x-5)=0\) (b) \(\left(x^{2}+3 x-1\right)+(2 x+3)(x-5) \geq 0\)

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$\text { 2; } P(x)=x^{2}+2 x-8$$

Solve each equation and inequality. (a) \(3(x+1)^{2}(2 x-1)^{4}+8(x+1)^{3}(2 x-1)^{3}=0\) (b) \(3(x+1)^{2}(2 x-1)^{4}+8(x+1)^{3}(2 x-1)^{3} \geq 0\)

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$-1: P(x)=x^{2}+4 x-5$$

Solve each equation and inequality. (a) \(4 x\left(x^{2}+1\right)\left(x^{2}+4\right)^{3}+6 x\left(x^{2}+1\right)^{2}\left(x^{2}+4\right)^{2}=0\) (b) \(4 x\left(x^{2}+1\right)\left(x^{2}+4\right)^{3}+6 x\left(x^{2}+1\right)^{2}\left(x^{2}+4\right)^{2}<0\)

Use the concepts of this section to answer the following. Show analytically that \(-2\) is a zero of multiplicity 2 of \(P(x)=x^{4}+2 x^{3}-7 x^{2}-20 x-12,\) and find all complex zeros. Then write \(P(x)\) in factored form.

Solve each equation and inequality, where \(k\) is a positive constant. (a) \(3 k x^{2}-7 x=0\) (b) \(3 k x^{2}-7 x<0\)

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$-4 ; \quad P(x)=9 x^{3}+39 x^{2}+12 x$$