Chapter 3

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. $$P(x)=x^{6}+16 x^{3}+64$$

Evaluate the expression and write the result in the form \(a+b i\) $$i^{1002}$$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=2 x^{3}-3 x^{2}-2 x+3$$

Find the maximum or minimum value of the function. $$f(x)=1+3 x-x^{2}$$

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$P(x)=x^{3}+3 x^{2}-4 x-12$$

Find the quotient and remainder using synthetic division. $$\frac{x^{3}-9 x^{2}+27 x-27}{x-3}$$

Find a polynomial with integer coefficients that satisfies the given conditions. \(P\) has degree 2 and zeros \(1+i\) and \(1-i\).

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{1}{i}$$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=4 x^{3}-7 x+3$$

Find the maximum or minimum value of the function. $$f(t)=100-49 t-7 t^{2}$$

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$P(x)=2 x^{3}-x^{2}-18 x+9$$

Find the quotient and remainder using synthetic division. $$\frac{2 x^{3}+3 x^{2}-2 x+1}{x-\frac{1}{2}}$$

Find a polynomial with integer coefficients that satisfies the given conditions. \(P\) has degree 2 and zeros \(1+i \sqrt{2}\) and \(1-i \sqrt{2}\).

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{1}{1+i}$$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=8 x^{3}+10 x^{2}-x-3$$

Find the maximum or minimum value of the function. $$f(t)=10 t^{2}+40 t+113$$

Find the quotient and remainder using synthetic division. $$\frac{6 x^{4}+10 x^{3}+5 x^{2}+x+1}{x+\frac{2}{3}}$$

Find a polynomial with integer coefficients that satisfies the given conditions. \(Q\) has degree 3 and zeros \(3,2 i\), and \(-2 i\).

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{2-3 i}{1-2 i}$$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=4 x^{3}+8 x^{2}-11 x-15$$

Find the maximum or minimum value of the function. $$f(s)=s^{2}-1.2 s+16$$

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$P(x)=x^{4}-2 x^{3}-8 x+16$$

Find the quotient and remainder using synthetic division. $$\frac{x^{3}-27}{x-3}$$

Find a polynomial with integer coefficients that satisfies the given conditions. \(Q\) has degree 3 and zeros 0 and \(i\).

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{5-i}{3+4 i}$$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=6 x^{3}+11 x^{2}-3 x-2$$

Find the maximum or minimum value of the function. $$g(x)=100 x^{2}-1500 x$$

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$P(x)=x^{4}-2 x^{3}+8 x-16$$

Find the quotient and remainder using synthetic division. $$\frac{x^{4}-16}{x+2}$$

Find a polynomial with integer coefficients that satisfies the given conditions. \(P\) has degree 3 and zeros 2 and \(i\).

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{26+39 i}{2-3 i}$$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=20 x^{3}-8 x^{2}-5 x+2$$

Find the maximum or minimum value of the function. $$h(x)=\frac{1}{2} x^{2}+2 x-6$$

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$P(x)=x^{4}-3 x^{2}-4$$

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=4 x^{2}+12 x+5, \quad c=-1$$

Find a polynomial with integer coefficients that satisfies the given conditions. \(Q\) has degree 3 and zeros \(-3\) and \(1+i\).

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{25}{4-3 i}$$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=12 x^{3}-20 x^{2}+x+3$$

Find the maximum or minimum value of the function. $$f(x)=-\frac{x^{2}}{3}+2 x+7$$

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$P(x)=x^{6}-2 x^{3}+1$$

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=2 x^{2}+9 x+1, \quad c=\frac{1}{2}$$

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{10 i}{1-2 i}$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$r(x)=\frac{4 x-4}{x+2}$$

Find a polynomial with integer coefficients that satisfies the given conditions. \(R\) has degree 4 and zeros \(1-2 i\) and \(1,\) with 1 a zero of multiplicity 2.

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=2 x^{4}-7 x^{3}+3 x^{2}+8 x-4$$

Find the maximum or minimum value of the function. $$f(x)=3-x-\frac{1}{2} x^{2}$$

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=x^{3}+3 x^{2}-7 x+6, \quad c=2$$

Evaluate the expression and write the result in the form \(a+b i\) $$(2-3 i)^{-1}$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$r(x)=\frac{2 x+6}{-6 x+3}$$

Find a polynomial with integer coefficients that satisfies the given conditions. \(S\) has degree 4 and zeros \(2 i\) and \(3 i\).