Chapter 3

A polynomial function \(f\) with real coefficients has the given degree, zeros, and solution point. Write the function (a) in completely factored form and (b) in polynomial form. Degree 3 Zeros $$-1,2+\sqrt{5} i$$ Solution Point $$f(2)=45$$

Use a graphing utility to graph the quadratic function and find the \(x\) -intercepts of the graph. Then find the \(x\) -intercepts algebraically to verify your answer. \(y=-\frac{1}{2}\left(x^{2}-6 x-7\right)\)

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(f(x)=x^{2}+x-2\)

When the graph of a rational function \(f\) has a vertical asymptote at \(x=4,\) can \(f\) have a common factor of \((x-4)\) in the numerator and denominator? Explain.

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, and slant asymptotes. $$f(x)=\frac{x^{3}}{x^{2}+4}$$

A polynomial function \(f\) with real coefficients has the given degree, zeros, and solution point. Write the function (a) in completely factored form and (b) in polynomial form. Degree 3 Zeros $$-2,2+2 \sqrt{2} i$$ Solution Point $$f(-1)=-34$$

Use a graphing utility to graph the quadratic function and find the \(x\) -intercepts of the graph. Then find the \(x\) -intercepts algebraically to verify your answer. \(y=\frac{7}{10}\left(x^{2}+12 x-45\right)\)

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(f(x)=2 x^{2}-14 x+24\)

Use a graphing utility to compare the graphs of \(y_{1}\) and \(y_{2}.\) $$y_{1}=\frac{3 x^{3}-5 x^{2}+4 x-5}{2 x^{2}-6 x+7}, \quad y_{2}=\frac{3 x^{3}}{2 x^{2}}$$ Start with a viewing window of \(-5 \leq x \leq 5\) and \(-10 \leq y \leq 10,\) and then zoom out. Make a conjecture about how the graph of a rational function \(f\) is related to the graph of \(y=a_{n} x^{n} / b_{m} x^{m},\) where \(a_{n} x^{n}\) is the leading term of the numerator of \(f\) and \(b_{m} x^{m}\) is the leading term of the denominator of \(f.\)

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, and slant asymptotes. $$f(x)=\frac{x^{3}+2 x^{2}+4}{2 x^{2}+1}$$

(a) verify the given factor(s) of the function \(f,\) (b) find the remaining factors of \(f,(\mathrm{c})\) use your results to write the complete factorization of \(f,\) and (d) list all real zeros of \(f .\) Confirm your results by using a graphing utility to graph the function. Factor(s) \((x+2)\) \((x+3)\) \((x-5),(x+4)\) \((x+2),(x-4)\) \((2 x+1)\) \((2 x-1)\) Function $$\begin{aligned} f(x)=x^{4}-4 x^{3}-15 x^{2} \\ &+58 x-40 \end{aligned}$$

Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given \(x\) -intercepts. (There are many correct answers.) (-1,0),(3,0)

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(f(t)=t^{3}-4 t^{2}+4 t\)

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, and slant asymptotes. $$f(x)=\frac{2 x^{2}-5 x+5}{x-2}$$

(a) verify the given factor(s) of the function \(f,\) (b) find the remaining factors of \(f,(\mathrm{c})\) use your results to write the complete factorization of \(f,\) and (d) list all real zeros of \(f .\) Confirm your results by using a graphing utility to graph the function. Factor(s) \((x+2)\) \((x+3)\) \((x-5),(x+4)\) \((x+2),(x-4)\) \((2 x+1)\) \((2 x-1)\) Function $$\begin{aligned} f(x)=8 x^{4}-14 x^{3}-71 x^{2} \\ &-10 x+24 \end{aligned}$$

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. $$f(x)=x^{4}+6 x^{2}-27$$

Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given \(x\) -intercepts. (There are many correct answers.) (0, 0), (10, 0)

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(f(x)=x^{4}-x^{3}-20 x^{2}\)

Write the general form of the equation of the line that passes through the points. $$(3,2),(0,-1)$$

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-2 x^{3}-3 x^{2}+12 x-18\) (Hint: One factor is \(x^{2}-6 .\) )

Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given \(x\) -intercepts. (There are many correct answers.) \((-3,0),\left(-\frac{1}{2}, 0\right)\)

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(f(x)=\frac{1}{2} x^{2}+\frac{5}{2} x-\frac{3}{2}\)

Write the general form of the equation of the line that passes through the points. $$(-6,1),(4,-5)$$

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-3 x^{3}-x^{2}-12 x-20\) (Hint: One factor is \(x^{2}+4 .\) )

Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given \(x\) -intercepts. (There are many correct answers.) \(\left(-\frac{5}{2}, 0\right),(2,0)\)

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(f(x)=\frac{5}{3} x^{2}+\frac{8}{3} x-\frac{4}{3}\)

Write the general form of the equation of the line that passes through the points. $$(2,7),(3,10)$$

Use the Rational Zero Test to list all possible rational zeros of \(f .\) Then find the rational zeros. $$f(x)=x^{3}+3 x^{2}-x-3$$

Use the given zero to find all the zeros of the function. Function $$f(x)=2 x^{3}+3 x^{2}+50 x+75$$ Zero $$5 i$$

Find the two positive real numbers with the given sum whose product is a maximum. The sum is \(110 .\)

Use a graphing utility to graph the function and approximate (accurate to three decimal places) any real zeros and relative extrema. \(f(x)=2 x^{4}-6 x^{2}+1\)

Write the general form of the equation of the line that passes through the points. $$(0,0),(-9,4)$$

Use the Rational Zero Test to list all possible rational zeros of \(f .\) Then find the rational zeros. $$f(x)=x^{3}-4 x^{2}-4 x+16$$

Use the given zero to find all the zeros of the function. Function $$f(x)=x^{3}+x^{2}+9 x+9$$ Zero $$3 i$$

Find the two positive real numbers with the given sum whose product is a maximum. The sum is 66

Use a graphing utility to graph the function and approximate (accurate to three decimal places) any real zeros and relative extrema. \(f(x)=-\frac{3}{8} x^{4}-x^{3}+2 x^{2}+5\)

Divide using long division. $$\left(x^{2}+5 x+6\right) \div(x-4)$$

Use a graphing utility to graph the rational function. Determine the domain of the function and identify any asymptotes. $$y=\frac{2 x^{2}+x}{x+1}$$

Use the Rational Zero Test to list all possible rational zeros of \(f .\) Then find the rational zeros. $$f(x)=2 x^{4}-17 x^{3}+35 x^{2}+9 x-45$$

Use the given zero to find all the zeros of the function. Function $$g(x)=x^{3}-7 x^{2}-x+87$$ Zero $$5+2 i$$

Use a graphing utility to graph the function and approximate (accurate to three decimal places) any real zeros and relative extrema. \(f(x)=x^{5}+3 x^{3}-x+6\)

Divide using long division. $$\left(x^{2}-10 x+15\right) \div(x-3)$$

Use a graphing utility to graph the rational function. Determine the domain of the function and identify any asymptotes. $$y=\frac{x^{2}+5 x+8}{x+3}$$

Use the Rational Zero Test to list all possible rational zeros of \(f .\) Then find the rational zeros. $$f(x)=4 x^{5}-8 x^{4}-5 x^{3}+10 x^{2}+x-2$$

Use the given zero to find all the zeros of the function. Function $$g(x)=4 x^{3}+23 x^{2}+34 x-10$$ Zero $$-3+i$$

Use a graphing utility to graph the function and approximate (accurate to three decimal places) any real zeros and relative extrema. \(f(x)=-3 x^{3}-4 x^{2}+x-3\)

Divide using long division. $$\left(2 x^{4}+x^{2}-11\right) \div\left(x^{2}+5\right)$$

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=2 x^{4}-x^{3}+6 x^{2}-x+5$$

Use a graphing utility to graph the rational function. Determine the domain of the function and identify any asymptotes. $$y=\frac{1+3 x^{2}-x^{3}}{x^{2}}$$

Use the given zero to find all the zeros of the function. Function $$h(x)=3 x^{3}-4 x^{2}+8 x+8$$ Zero $$1-\sqrt{3} i$$