Chapter 3

For each polynomial function, find (a) the end behavior; (b) the \(y\) -intercept; (c) the \(x\) -intercept(s) of the graph of the function and the multiplicities of the real zeros; (d) the symmetries of the graph of the function, if any; and (e) the intervals on which the function is positive or negative. Use this information to sketch a graph of the function. Factor first if the expression is not in factored form. $$f(x)=(x-1)(x+2)^{2}(x+1)$$

Solve the rational inequality. $$\frac{4-x}{x-1}>x$$

Find all the real zeros of the polynomial. $$h(x)=x^{4}+3 x^{3}-8 x^{2}-22 x-24$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{-12}{x+6}$$

One zero of each polynomial is given. Use it to express the polynomial as a product of linear factors over the complex numbers. You may have already factored some of these polynomials into linear and irreducible quadratic factors in the previous group of exercises. $$2 x^{3}-9 x^{2}-11 x+30 ; \text { zero: } x=5$$

Determine together \(q(x)\) is a factor of \(p(x)\) Here, \(p(x)\) is the first polynomial and \(q(x)\) is the second polynomial. justify your answer. $$x^{5}-3 x^{3}+2 x-8 ; x-4$$

Sketch the polynomial function using transformations. $$f(x)=x^{3}-2$$

For each polynomial function, find (a) the end behavior; (b) the \(y\) -intercept; (c) the \(x\) -intercept(s) of the graph of the function and the multiplicities of the real zeros; (d) the symmetries of the graph of the function, if any; and (e) the intervals on which the function is positive or negative. Use this information to sketch a graph of the function. Factor first if the expression is not in factored form. $$g(x)=-2(x+1)^{2}(x-3)^{2}$$

Solve the rational inequality. $$\frac{-8}{x+3}<-2 x$$

Find all the real zeros of the polynomial. $$f(x)=x^{5}-7 x^{4}+10 x^{3}+14 x^{2}-24 x$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{-10}{x+2}$$

One zero of each polynomial is given. Use it to express the polynomial as a product of linear factors over the complex numbers. You may have already factored some of these polynomials into linear and irreducible quadratic factors in the previous group of exercises. $$2 x^{3}-9 x^{2}+7 x+6 ; \text { zero: } x=2$$

Determine together \(q(x)\) is a factor of \(p(x)\) Here, \(p(x)\) is the first polynomial and \(q(x)\) is the second polynomial. justify your answer. $$-2 x^{4}-7 x^{3}+5 ; x+2$$

Sketch the polynomial function using transformations. $$f(x)=x^{4}-1$$

For each polynomial function, find (a) the end behavior; (b) the \(y\) -intercept; (c) the \(x\) -intercept(s) of the graph of the function and the multiplicities of the real zeros; (d) the symmetries of the graph of the function, if any; and (e) the intervals on which the function is positive or negative. Use this information to sketch a graph of the function. Factor first if the expression is not in factored form. $$f(x)=-3(x-2)^{2}(x+1)^{2}$$

Solve the rational inequality. $$\frac{1}{x} \leq \frac{1}{2 x-1}$$

Find all real solutions of the polynomial equation. $$x^{3}+2 x^{2}+2 x=-1$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{12}{3-x}$$

One zero of each polynomial is given. Use it to express the polynomial as a product of linear factors over the complex numbers. You may have already factored some of these polynomials into linear and irreducible quadratic factors in the previous group of exercises. $$x^{4}-5 x^{3}+7 x^{2}-5 x+6 ; \text { zero: } x=2$$

Determine together \(q(x)\) is a factor of \(p(x)\) Here, \(p(x)\) is the first polynomial and \(q(x)\) is the second polynomial. justify your answer. $$x^{4}-50 ; x-5$$

Sketch the polynomial function using transformations. $$f(x)=\frac{1}{2} x^{3}$$

For each polynomial function, find (a) the end behavior; (b) the \(y\) -intercept; (c) the \(x\) -intercept(s) of the graph of the function and the multiplicities of the real zeros; (d) the symmetries of the graph of the function, if any; and (e) the intervals on which the function is positive or negative. Use this information to sketch a graph of the function. Factor first if the expression is not in factored form. $$f(x)=x^{3}+4 x^{2}+4 x$$

Solve the rational inequality. $$\frac{2}{x+1}>\frac{1}{x-2}$$

Find all real solutions of the polynomial equation. $$3 x^{3}-7 x^{2}=-5 x+1$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{8}{4-x}$$

Determine together \(q(x)\) is a factor of \(p(x)\) Here, \(p(x)\) is the first polynomial and \(q(x)\) is the second polynomial. justify your answer. $$2 x^{5}-1 ; x-2$$

Sketch the polynomial function using transformations. $$g(x)=-\frac{1}{2} x^{4}$$

For each polynomial function, find (a) the end behavior; (b) the \(y\) -intercept; (c) the \(x\) -intercept(s) of the graph of the function and the multiplicities of the real zeros; (d) the symmetries of the graph of the function, if any; and (e) the intervals on which the function is positive or negative. Use this information to sketch a graph of the function. Factor first if the expression is not in factored form. $$f(x)=-x^{3}-2 x^{2}-x$$

Solve the rational inequality. $$\frac{3}{x-1} \leq 2$$

Find all real solutions of the polynomial equation. $$x^{3}-6 x^{2}+5 x=-12$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{3}{(x+1)^{2}}$$

One zero of each polynomial is given. Use it to express the polynomial as a product of linear factors over the complex numbers. You may have already factored some of these polynomials into linear and irreducible quadratic factors in the previous group of exercises. $$x^{4}+4 x^{3}-x^{2}+16 x-20 ; \text { zero: } x=-5$$

Determine together \(q(x)\) is a factor of \(p(x)\) Here, \(p(x)\) is the first polynomial and \(q(x)\) is the second polynomial. justify your answer. $$3 x^{3}-48 x-4 x^{2}+64 ; x+4$$

Sketch the polynomial function using transformations. $$g(x)=(x-2)^{3}$$

For each polynomial function, find (a) the end behavior; (b) the \(y\) -intercept; (c) the \(x\) -intercept(s) of the graph of the function and the multiplicities of the real zeros; (d) the symmetries of the graph of the function, if any; and (e) the intervals on which the function is positive or negative. Use this information to sketch a graph of the function. Factor first if the expression is not in factored form. $$h(x)=-2 x^{4}+4 x^{3}+2 x^{2}$$

Solve the rational inequality. $$\frac{-1}{2 x+1} \geq 1$$

Find all real solutions of the polynomial equation. $$4 x^{3}-16 x^{2}+19 x=-6$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$h(x)=\frac{-9}{(x-3)^{2}}$$

One zero of each polynomial is given. Use it to express the polynomial as a product of linear factors over the complex numbers. You may have already factored some of these polynomials into linear and irreducible quadratic factors in the previous group of exercises. $$x^{4}-6 x^{3}+9 x^{2}-24 x+20 ; \text { zero: } x=5$$

Determine together \(q(x)\) is a factor of \(p(x)\) Here, \(p(x)\) is the first polynomial and \(q(x)\) is the second polynomial. justify your answer. $$x^{3}+9 x+x^{2}+9 ; x+1$$

Sketch the polynomial function using transformations. $$h(x)=(x+1)^{4}$$

For each polynomial function, find (a) the end behavior; (b) the \(y\) -intercept; (c) the \(x\) -intercept(s) of the graph of the function and the multiplicities of the real zeros; (d) the symmetries of the graph of the function, if any; and (e) the intervals on which the function is positive or negative. Use this information to sketch a graph of the function. Factor first if the expression is not in factored form. $$f(x)=3 x^{4}-6 x^{3}+3 x^{2}$$

Solve the rational inequality. $$\frac{x-1}{x+2} \geq 0$$

Find all real solutions of the polynomial equation. $$2 x^{3}-3 x^{2}=11 x-6$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$g(x)=\frac{3-x}{x+4}$$

Find an expression for a polynomial \(p(x)\) with real coefficients that satisfies the given conditions. There may be more than one possible answer. Degree \(2 ; x=2\) and \(x=-1\) are zeros

Sketch the polynomial function using transformations. $$h(x)=-2 x^{5}-1$$

Find an expression for a polynomial function \(f(x)\) having the given properties. There can be more than one correct anstoer. Degrec \(3 ;\) zeros \(-2,5,\) and \(6,\) cach of multiplicity 1

Solve the rational inequality. $$\frac{3 x+6}{x-3}<0$$

Find all real solutions of the polynomial equation. $$2 x^{3}-x^{2}-18 x=-9$$