Chapter 3

In Problems \(1-10\), use long division to find the quotient \(q(x)\) and remainder \(r(x)\) when the polynomial \(f(x)\) is divided by the given polynomial \(d(x)\). In each case write your answer in the form \(f(x)=\) \(d(x) q(x)+r(x)\). $$ f(x)=8 x^{2}+4 x-7 ; d(x)=x^{2} $$

In Problems \(1-8,\) proceed as in Example 1 and use transformations to sketch the graph of the given polynomial function. \(y=x^{3}-3\)

In Problems 1 and 2 , use a calculator to fill out the given table for the rational \(f(x)=\frac{2 x}{x-3}\). \(x=3\) is a vertical asymptote for the graph of \(f\) $$ \begin{array}{|c|c|c|c|c|c|} \hline x & 3.1 & 3.01 & 3.001 & 3.0001 & 3.00001 \\ \hline f(x) & & & & & \\ \hline x & 2.9 & 2.99 & 2.999 & 2.9999 & 2.99999 \\ \hline f(x) & & & & & \\ \hline \end{array} $$

In Problems \(1-20,\) find all rational zeros of the given polynomial function \(f\). $$ f(x)=5 x^{3}-3 x^{2}+8 x+4 $$

Determine whether the indicated real number is a zero of the given polynomial function \(f .\) If yes, find all other zeros and then give the complete factorization of \(f(x)\). $$ 1 ; f(x)=4 x^{3}-9 x^{2}+6 x-1 $$

Use long division to find the quotient \(q(x)\) and remainder \(r(x)\) when the polynomial \(f(x)\) is divided by the given polynomial \(d(x)\). In each case write your answer in the form \(f(x)=\) \(d(x) q(x)+r(x)\). $$ f(x)=x^{2}+2 x-3 ; d(x)=x^{2}+1 $$

Use a calculator to fill out the given table for the rational \(f(x)=\frac{2 x}{x-3}\). \(y=2\) is a horizontal asymptote for the graph of \(f\) $$ \begin{array}{|c|c|c|c|c|c|} \hline x & 10 & 100 & 1000 & 10,000 & 100,000 \\ \hline f(x) & & & & & \\ \hline x & -10 & -100 & -1000 & -10,000 & -100,000 \\ \hline f(x) & & & & & \\ \hline \end{array} $$

Find an approximation that is accurate to three decimal places to the zero of \(f(x)=x^{3}\) \(-3 x-1\) in the given interval. [-1,0]

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=2 x^{3}+3 x^{2}-x+2 $$

Determine whether the indicated real number is a zero of the given polynomial function \(f .\) If yes, find all other zeros and then give the complete factorization of \(f(x)\). $$ \frac{1}{2} ; f(x)=2 x^{3}-x^{2}+32 x-16 $$

Use long division to find the quotient \(q(x)\) and remainder \(r(x)\) when the polynomial \(f(x)\) is divided by the given polynomial \(d(x)\). In each case write your answer in the form \(f(x)=\) \(d(x) q(x)+r(x)\). $$ f(x)=5 x^{3}-7 x^{2}+4 x+1 ; d(x)=x^{2}+x-1 $$

Proceed as in Example 1 and use transformations to sketch the graph of the given polynomial function. \(y=(x-2)^{3}+2\)

In Problems 3-22, find the vertical and horizontal asymptotes for the graph of the given rational function. Find \(x\) - and \(y\) -intercepts of the graph. Sketch the graph of \(f\). $$ f(x)=\frac{1}{x-2} $$

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=x^{3}-8 x-3 $$

Determine whether the indicated real number is a zero of the given polynomial function \(f .\) If yes, find all other zeros and then give the complete factorization of \(f(x)\). $$ 5 ; f(x)=x^{3}-6 x^{2}+6 x+5 $$

Use long division to find the quotient \(q(x)\) and remainder \(r(x)\) when the polynomial \(f(x)\) is divided by the given polynomial \(d(x)\). In each case write your answer in the form \(f(x)=\) \(d(x) q(x)+r(x)\). $$ f(x)=14 x^{3}-12 x^{2}+6 ; d(x)=x^{2}-1 $$

Proceed as in Example 1 and use transformations to sketch the graph of the given polynomial function. \(y=3-(x+2)^{3}\)

Find the vertical and horizontal asymptotes for the graph of the given rational function. Find \(x\) - and \(y\) -intercepts of the graph. Sketch the graph of \(f\). $$ f(x)=\frac{4}{x+3} $$

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=2 x^{3}-7 x^{2}-17 x+10 $$

Determine whether the indicated real number is a zero of the given polynomial function \(f .\) If yes, find all other zeros and then give the complete factorization of \(f(x)\). $$ 3 ; f(x)=x^{3}-3 x^{2}+4 x-12 $$

Use long division to find the quotient \(q(x)\) and remainder \(r(x)\) when the polynomial \(f(x)\) is divided by the given polynomial \(d(x)\). In each case write your answer in the form \(f(x)=\) \(d(x) q(x)+r(x)\). $$ f(x)=2 x^{3}+4 x^{2}-3 x+5 ; d(x)=(x+2)^{2} $$

Approximate the area under the graph of \(f(x)=x\) +2 on the interval [-1,2] using six subintervals of equal width and choosing: (a) \(x_{k}^{*}\) as the left-hand endpoint of each subinterval, and (b) \(x_{k}^{m}\) as the right-hand endpoint of each subinterval.

Find the vertical and horizontal asymptotes for the graph of the given rational function. Find \(x\) - and \(y\) -intercepts of the graph. Sketch the graph of \(f\). $$ f(x)=\frac{x}{x+1} $$

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=4 x^{4}-7 x^{2}+5 x-1 $$

Determine whether the indicated real number is a zero of the given polynomial function \(f .\) If yes, find all other zeros and then give the complete factorization of \(f(x)\). $$ -\frac{2}{3} ; f(x)=3 x^{3}-10 x^{2}-2 x+4 $$

Use long division to find the quotient \(q(x)\) and remainder \(r(x)\) when the polynomial \(f(x)\) is divided by the given polynomial \(d(x)\). In each case write your answer in the form \(f(x)=\) \(d(x) q(x)+r(x)\). $$ f(x)=x^{3}+x^{2}+x+1 ; d(x)=(2 x+1)^{2} $$

Find the vertical and horizontal asymptotes for the graph of the given rational function. Find \(x\) - and \(y\) -intercepts of the graph. Sketch the graph of \(f\). $$ f(x)=\frac{x}{2 x-5} $$

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=8 x^{4}-2 x^{3}+15 x^{2}-4 x-2 $$

Determine whether the indicated real number is a zero of the given polynomial function \(f .\) If yes, find all other zeros and then give the complete factorization of \(f(x)\). $$ -2 ; f(x)=x^{3}-4 x^{2}-2 x+20 $$

Use long division to find the quotient \(q(x)\) and remainder \(r(x)\) when the polynomial \(f(x)\) is divided by the given polynomial \(d(x)\). In each case write your answer in the form \(f(x)=\) \(d(x) q(x)+r(x)\). $$ f(x)=27 x^{3}+x-2 ; d(x)=3 x^{2}-x $$

Proceed as in Example 1 and use transformations to sketch the graph of the given polynomial function. \(y=1-(x-1)^{4}\)

Approximate the area under the graph of \(f(x)=-\) \(x^{2}+5 x\) on the interval [0,5] using five subintervals of equal width and choosing: (a) \(x_{k}^{\text {as the left-hand endpoint of each }}\) subinterval, and (b) \(x_{k}^{\text {as the right-hand endpoint of each }}\) subinterval.

Find the vertical and horizontal asymptotes for the graph of the given rational function. Find \(x\) - and \(y\) -intercepts of the graph. Sketch the graph of \(f\). $$ f(x)=\frac{4 x-9}{2 x+3} $$

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=x^{4}+2 x^{3}+10 x^{2}+14 x+21 $$

Verify that each of the indicated numbers are zeros of the given polynomial function \(f\). Find all other zeros and then give the complete factorization of \(f(x)\) $$ -3,5 ; f(x)=4 x^{4}-8 x^{3}-61 x^{2}+2 x+15 $$

Use long division to find the quotient \(q(x)\) and remainder \(r(x)\) when the polynomial \(f(x)\) is divided by the given polynomial \(d(x)\). In each case write your answer in the form \(f(x)=\) \(d(x) q(x)+r(x)\). $$ f(x)=x^{4}+8 ; d(x)=x^{3}+2 x-1 $$

Proceed as in Example 1 and use transformations to sketch the graph of the given polynomial function. \(y=4+(x+1)^{4}\)

Find the vertical and horizontal asymptotes for the graph of the given rational function. Find \(x\) - and \(y\) -intercepts of the graph. Sketch the graph of \(f\). $$ f(x)=\frac{2 x+4}{x-2} $$

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=3 x^{4}+5 x^{2}+1 $$

Verify that each of the indicated numbers are zeros of the given polynomial function \(f\). Find all other zeros and then give the complete factorization of \(f(x)\) $$ \frac{1}{4}, \frac{3}{2} ; f(x)=8 x^{4}-30 x^{3}+23 x^{2}+8 x-3 $$

Use long division to find the quotient \(q(x)\) and remainder \(r(x)\) when the polynomial \(f(x)\) is divided by the given polynomial \(d(x)\). In each case write your answer in the form \(f(x)=\) \(d(x) q(x)+r(x)\). $$ f(x)=6 x^{5}+4 x^{4}+x^{3} ; d(x)=x^{3}-2 $$

In Problems 9-12, determine whether the given polynomial function \(f\) is even, odd, or neither even nor odd. Do not graph. \(f(x)=-2 x^{3}+4 x\)

Approximate the area under the graph of \(f(x)=-\) \(x^{2}+5 x\) on the interval [0,5] using five subintervals of equal width and choosing \(x_{k}\) as the midpoints of each subinterval.

Find the vertical and horizontal asymptotes for the graph of the given rational function. Find \(x\) - and \(y\) -intercepts of the graph. Sketch the graph of \(f\). $$ f(x)=\frac{1-x}{x+1} $$

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=6 x^{4}-5 x^{3}-2 x^{2}-8 x+3 $$

Verify that each of the indicated numbers are zeros of the given polynomial function \(f\). Find all other zeros and then give the complete factorization of \(f(x)\) $$ \begin{aligned} &1,-\frac{1}{3}(\text { multiplicity } 2) ; f(x)=9 x^{4}+69 x^{3}-29 x^{2}-41 x-8 \end{aligned} $$

Use long division to find the quotient \(q(x)\) and remainder \(r(x)\) when the polynomial \(f(x)\) is divided by the given polynomial \(d(x)\). In each case write your answer in the form \(f(x)=\) \(d(x) q(x)+r(x)\). $$ \begin{array}{l} f(x)=5 x^{6}-x^{5}+10 x^{4}+3 x^{2}-2 x+4 ; d(x)=x^{2}+x \\ -1 \end{array} $$

Determine whether the given polynomial function \(f\) is even, odd, or neither even nor odd. Do not graph. \(f(x)=x^{6}-5 x^{2}+7\)

Approximate the area under the graph of \(f(x)=-\) \(x^{3}+2 x^{2}\) on the interval [0,2] using ten subintervals of equal width and choosing \(x_{k}^{*}\) as the right-hand endpoint of each subinterval.

Find the vertical and horizontal asymptotes for the graph of the given rational function. Find \(x\) - and \(y\) -intercepts of the graph. Sketch the graph of \(f\). $$ f(x)=\frac{2 x-3}{x} $$