Chapter 4

\(9-14\) . Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$ \frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)} $$ $$ P(x)=6 x^{3}+x^{2}-12 x+5, \quad D(x)=3 x-4 $$

A polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{4}+6 x^{2}+9\)

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). (c) Sketch its graph. $$ f(x)=-x^{2}+10 x $$

Find the \(x\) -and \(y\) -intercepts of the rational function. $$ t(x)=\frac{x^{2}-x-2}{x-6} $$

\(9-14\) . Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$ \frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)} $$ $$ P(x)=2 x^{4}-x^{3}+9 x^{2}, \quad D(x)=x^{2}+4 $$

A polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{3}+8\)

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). (c) Sketch its graph. $$ f(x)=x^{2}+4 x+3 $$

Find the \(x\) -and \(y\) -intercepts of the rational function. $$ r(x)=\frac{2}{x^{2}+3 x-4} $$

\(9-14\) . Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$ \frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)} $$ $$ P(x)=x^{5}+x^{4}-2 x^{3}+x+1, \quad D(x)=x^{2}+x-1 $$

A polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{3}-8\)

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). (c) Sketch its graph. $$ f(x)=x^{2}-2 x+2 $$

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

Find the \(x\) -and \(y\) -intercepts of the rational function. $$ r(x)=\frac{x^{2}-9}{x^{2}} $$

\(15-24=\) Find the quotient and remainder using long division. $$ \frac{x^{2}-6 x-8}{x-4} $$

A polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{6}-1\)

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). (c) Sketch its graph. $$ f(x)=-x^{2}+6 x+4 $$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=x^{3}-7 x^{2}+14 x-8 $$

Find the \(x\) -and \(y\) -intercepts of the rational function. $$ r(x)=\frac{x^{3}+8}{x^{2}+4} $$

\(15-24=\) Find the quotient and remainder using long division. $$\frac{x^{3}-x^{2}-2 x+6}{x-2}$$

A polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{6}-7 x^{3}-8\)

\(15-26\) Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=(x-1)(x+1)(x-2) $$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). (c) Sketch its graph. $$ f(x)=-x^{2}-4 x+4 $$

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

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. \(P(x)=x^{2}+25\)

\(15-26\) Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=x(x-3)(x+2) $$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). (c) Sketch its graph. $$ f(x)=2 x^{2}+4 x+3 $$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=x^{3}+4 x^{2}-3 x-18 $$

\(15-24=\) Find the quotient and remainder using long division. $$ \frac{x^{3}+3 x^{2}+4 x+3}{3 x+6} $$

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. \(P(x)=4 x^{2}+9\)

\(15-26\) Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=(2 x-1)(x+1)(x+3) $$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). (c) Sketch its graph. $$ f(x)=-3 x^{2}+6 x-2 $$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=x^{3}-6 x^{2}+12 x-8 $$

\(15-24=\) Find the quotient and remainder using long division. $$ \frac{x^{3}+6 x+3}{x^{2}-2 x+2} $$

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

\(15-26\) Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=(x-3)(x+2)(3 x-2) $$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). (c) Sketch its graph. $$ f(x)=2 x^{2}-20 x+57 $$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=x^{3}-x^{2}-8 x+12 $$

\(15-24=\) Find the quotient and remainder using long division. $$ \frac{3 x^{4}-5 x^{3}-20 x-5}{x^{2}+x+3} $$

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. \(Q(x)=x^{2}-8 x+17\)

\(15-26\) Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=\frac{1}{5} x(x-5)^{2} $$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). (c) Sketch its graph. $$ f(x)=2 x^{2}+x-6 $$

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

Find all horizontal and vertical asymptotes (if any). $$ r(x)=\frac{5}{x-2} $$

\(15-24=\) Find the quotient and remainder using long division. $$ \frac{6 x^{3}+2 x^{2}+22 x}{2 x^{2}+5} $$

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

\(15-26\) Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=(x-1)^{2}(x-3) $$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). (c) Sketch its graph. $$ f(x)=-4 x^{2}-16 x+3 $$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=x^{3}-4 x^{2}-7 x+10 $$

Find all horizontal and vertical asymptotes (if any). $$ r(x)=\frac{2 x-3}{x^{2}-1} $$

\(15-24=\) Find the quotient and remainder using long division. $$ \frac{9 x^{2}-x+5}{3 x^{2}-7 x} $$