Chapter 4

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=3 x^{4}-10 x^{3}-9 x^{2}+40 x-12 $$

Find all horizontal and vertical asymptotes (if any). $$ t(x)=\frac{x^{2}+2}{x-1} $$

25-38 . Find the quotient and remainder using synthetic division. $$ \frac{x^{3}-8 x+2}{x+3} $$

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. \(P(x)=x^{4}+3 x^{2}-4\)

\(27-40\) Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=x^{4}-3 x^{3}+2 x^{2} $$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $$ h(x)=1-x-x^{2} $$

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

Find all horizontal and vertical asymptotes (if any). $$ r(x)=\frac{x^{3}+3 x^{2}}{x^{2}-4} $$

25-38 . Find the quotient and remainder using synthetic division. $$ \frac{x^{4}-x^{3}+x^{2}-x+2}{x-2} $$

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. \(P(x)=x^{5}+7 x^{3}\)

\(27-40\) Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=x^{5}-9 x^{3} $$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $$ h(x)=3-4 x-4 x^{2} $$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=4 x^{3}+4 x^{2}-x-1 $$

25-38 . Find the quotient and remainder using synthetic division. $$ \frac{x^{5}+3 x^{3}-6}{x-1} $$

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. \(P(x)=x^{5}+6 x^{3}+9 x\)

\(27-40\) Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=x^{3}+x^{2}-x-1 $$

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

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

25-38 . Find the quotient and remainder using synthetic division. $$ \frac{x^{3}-9 x^{2}+27 x-27}{x-3} $$

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

\(27-40\) 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 maximum or minimum value of the function. $$ f(x)=1+3 x-x^{2} $$

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

Use transformations of the graph of \(y=\frac{1}{x}\) to graph the rational function, as in Example \(2 .\) $$ s(x)=\frac{3}{x+1} $$

25-38 . 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 \(1+i\) and \(1-i\)

\(27-40\) 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 maximum or minimum value of the function. $$ f(t)=100-49 t-7 t^{2} $$

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

25-38 . 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. \(P\) has degree 2 and zeros \(1+i \sqrt{2}\) and \(1-i \sqrt{2}\)

\(27-40\) Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=\frac{1}{8}\left(2 x^{4}+3 x^{3}-16 x-24\right)^{2} $$

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

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

25-38 . 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 \(3,2 i,\) and \(-2 i\)

\(27-40\) 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 maximum or minimum value of the function. $$ f(s)=s^{2}-1.2 s+16 $$

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

25-38 . 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. \(Q\) has degree 3 and zeros 0 and \(i\)

\(27-40\) 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 maximum or minimum value of the function. $$ g(x)=100 x^{2}-1500 x $$

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

\(39-51\) . 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. \(P\) has degree 3 and zeros 2 and \(i\)

\(27-40\) Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=x^{4}-3 x^{2}-4 $$

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

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

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=2 x^{2}+9 x+1, \quad c=\frac{1}{2} $$