91Ó°ÊÓ

Determine the end behavior of the function. $$f(x)=3 x^{3}-4 x^{2}+5$$

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. $$g(x)=(x+1)(x-2)^{2}$$

Solve the rational inequality. $$\frac{4 x^{2}-9}{x+2}<0$$

Find all the real zeros of the polynomial. $$G(x)=2 x^{3}+x^{2}-16 x-15$$

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. $$x^{3}-5 x^{2}+8 x-4 ; x+2$$

Let \(f(x)=\frac{-3 x^{2}+4}{x^{2}}\) (a) Fill in the following table for values of \(x\) near zero. What do you observe about the value of \(f(x)\) as \(x\) approaches zero from the right? from the left? $$\begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & -0.5 & -0.1 & -0.01 & 0.01 & 0.1 & 0.05 \\ \hline f(\boldsymbol{x}) & & & & & \end{array}$$ (b) Complete the following table. What happens to the value of \(f(x)\) as \(x\) gets very large and positive? $$\begin{array}{|c|c|c|c|c|} \hline x & 10 & 50 & 100 & 1000 \\ \hline f(x) & & & & \\ \hline \end{array}$$ (c) Complete the following table. What happens to the value of \(f(x)\) as \(x\) gets very large and negative? (TABLE CAN'T COPY)

Determine the end behavior of the function. $$f(s)=\frac{7}{2} s^{5}-14 s^{3}+10 s$$

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

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. $$x^{3}-2 x^{2}+x-2 ; \text { zero: } x=2$$