Chapter 3

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=6 x^{4}-7 x^{3}-12 x^{2}+3 x+2$$

Find the maximum or minimum value of the function. $$g(x)=2 x(x-4)+7$$

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=x^{3}-x^{2}+x+5, \quad c=-1$$

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{4+6 i}{3 i}$$

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. $$s(x)=\frac{4-3 x}{x+7}$$

Find a polynomial with integer coefficients that satisfies the given conditions. \(T\) has degree \(4,\) zeros \(i\) and \(1+i,\) and constant term 12.

Find a function whose graph is a parabola with vertex \((1,-2)\) and that passes through the point \((4,16)\)

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

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{-3+5 i}{15 i}$$

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. $$s(x)=\frac{1-2 x}{2 x+3}$$

Find a polynomial with integer coefficients that satisfies the given conditions. \(U\) has degree \(5,\) zeros \(\underline{4},-1,\) and \(-i,\) and leading coefficient 4 the zero \(-1\) has multiplicity 2.

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=x^{5}-4 x^{4}-3 x^{3}+22 x^{2}-4 x-24$$

Find a function whose graph is a parabola with vertex \((3,4)\) and that passes through the point \((1,-8)\)

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=2 x^{3}-21 x^{2}+9 x-200, \quad c=11$$

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

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. $$r(x)=\frac{18}{(x-3)^{2}}$$

Find all zeros of the polynomial. $$P(x)=x^{3}+2 x^{2}+4 x+8$$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=3 x^{5}-14 x^{4}-14 x^{3}+36 x^{2}+43 x+10$$

Find the domain and range of the function. $$f(x)=-x^{2}+4 x-3$$

Determine the end behavior of \(P\). Compare the graphs of \(P\) and \(Q\) in large and small viewing rectangles, as in Example \(3(b)\). $$P(x)=x^{11}-9 x^{9} ; \quad Q(x)=x^{11}$$

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=5 x^{4}+30 x^{3}-40 x^{2}+36 x+14, \quad c=-7$$

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

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. $$r(x)=\frac{x-2}{(x+1)^{2}}$$

Find all zeros of the polynomial. $$P(x)=x^{3}-7 x^{2}+17 x-15$$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=2 x^{6}-3 x^{5}-13 x^{4}+29 x^{3}-27 x^{2}+32 x-12$$

Find the domain and range of the function. $$f(x)=x^{2}-2 x-3$$

Determine the end behavior of \(P\). Compare the graphs of \(P\) and \(Q\) in large and small viewing rectangles, as in Example \(3(b)\). $$P(x)=2 x^{2}-x^{12} ; \quad Q(x)=-x^{12}$$

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=6 x^{5}+10 x^{3}+x+1, \quad c=-2$$

Evaluate the radical expression and express the result in the form \(a+b i\) $$\sqrt{-25}$$

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. $$s(x)=\frac{4 x-8}{(x-4)(x+1)}$$

Find all zeros of the polynomial. $$P(x)=x^{3}-2 x^{2}+2 x-1$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\). $$P(x)=x^{3}+4 x^{2}+3 x-2$$

Find the domain and range of the function. $$f(x)=2 x^{2}+6 x-7$$

The graph of a polynomial function is given. From the graph, find (a) the \(x\) - and \(y\) -intercepts, and (b) the coordinates of all local extrema. $$P(x)=-x^{2}+4 x$$

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=x^{7}-3 x^{2}-1, \quad c=3$$

Evaluate the radical expression and express the result in the form \(a+b i\) $$\sqrt{\frac{-9}{4}}$$

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. $$s(x)=\frac{x+2}{(x+3)(x-1)}$$

Find all zeros of the polynomial. $$P(x)=x^{3}+7 x^{2}+18 x+18$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\). $$P(x)=x^{3}-5 x^{2}+2 x+12$$

Find the domain and range of the function. $$f(x)=-3 x^{2}+6 x+4$$

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=-2 x^{6}+7 x^{5}+40 x^{4}-7 x^{2}+10 x+112, \quad c=-3$$

Evaluate the radical expression and express the result in the form \(a+b i\) $$\sqrt{-3} \sqrt{-12}$$

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. $$s(x)=\frac{6}{x^{2}-5 x-6}$$

Find all zeros of the polynomial. $$P(x)=x^{3}-3 x^{2}+3 x-2$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\). $$P(x)=x^{4}-6 x^{3}+4 x^{2}+15 x+4$$

A quadratic function is given. (a) Use a graphing device to find the maximum or minimum value of the quadratic function \(f\) correct to two decimal places. (b) Find the exact maximum or minimum value of \(f,\) and compare it with your answer to part (a). $$f(x)=x^{2}+1.79 x-3.21$$

The graph of a polynomial function is given. From the graph, find (a) the \(x\) - and \(y\) -intercepts, and (b) the coordinates of all local extrema. $$P(x)=-\frac{1}{2} x^{3}+\frac{3}{2} x-1$$

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=3 x^{3}+4 x^{2}-2 x+1, \quad c=\frac{2}{3}$$

Evaluate the radical expression and express the result in the form \(a+b i\) $$\sqrt{\frac{1}{3}} \sqrt{-27}$$

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. $$s(x)=\frac{2 x-4}{x^{2}+x-2}$$