Chapter 6

Simplify. $$ 2 x(3 y+9) $$

Simplify. Assume that no variable equals 0. $$ \left(\frac{c d}{3}\right)^{-2} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{3}-3 x+2 ; x-1 $$

For Exercises \(7-10\) , use the following information. The number of cable TV systems after 1985 can be modeled by the function \(C(t)=-43.2 t^{2}+1343 t+790,\) where \(t\) represents the number of years since \(1985 .\) Graph this equation for the years 1985 to 2005 .

Find all of the zeros of each function. \(v(x)=x^{3}-3 x^{2}+4 x-12\)

Factor completely. If the polynomial is not factorable, write prime. $$ 16 w^{2}-169 $$

If \(p(x)=2 x^{3}+6 x-12\) and \(q(x)=5 x^{2}+4,\) find each value. 5\([q(2 a)]\)

Simplify. $$ \left(x^{3}+y^{3}\right) \div(x+y) $$

Simplify. $$ 2 p^{2} q\left(5 p q-3 p^{3} q^{2}+4 p q^{4}\right) $$

Simplify. Assume that no variable equals 0. $$ \left(n^{3}\right)^{3}\left(n^{-3}\right)^{3} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ 6 x^{3}-25 x^{2}+2 x+8 ; 3 x-2 $$

For Exercises \(7-10\) , use the following information. The number of cable TV systems after 1985 can be modeled by the function \(C(t)=-43.2 t^{2}+1343 t+790,\) where \(t\) represents the number of years since \(1985 .\) Describe the turning points of the graph and its end behavior.

Find all of the zeros of each function. \(f(x)=6 x^{3}+5 x^{2}-9 x+2\)

Find all of the zeros of each function. \(f(x)=x^{3}-3 x^{2}+9 x+13\)

Factor completely. If the polynomial is not factorable, write prime. $$ h^{3}+8000 $$

If \(p(x)=2 x^{3}+6 x-12\) and \(q(x)=5 x^{2}+4,\) find each value. \(3 p(a)-q(a+1)\)

Simplify. $$ \frac{x^{3}+13 x^{2}-12 x-8}{x+2} $$

Simplify. $$ (y-10)(y+7) $$

Simplify. Assume that no variable equals 0. $$ \frac{81 p^{6} q^{5}}{\left(3 p^{2} q\right)^{2}} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{4}+2 x^{3}-8 x-16 ; x+2 $$

Find all of the zeros of each function. \(f(x)=x^{4}-x^{3}-x^{2}-x-2\)

Write a polynomial function of least degree with integral coefficients the zeros of which include 2 and 4\(i .\)

Write each expression in quadratic form, if possible. $$ 5 y^{4}+7 y^{3}-8 $$

Simplify. $$ \left(b^{4}-2 b^{3}+b^{2}-3 b+4\right)(b-2)^{-1} $$

Simplify. $$ (x+6)(x+3) $$

Simplify. Assume that no variable equals 0. $$ \left(\frac{-6 x^{6}}{3 x^{3}}\right)^{-2} $$

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=x^{2}-8 x+6 $$

List all of the possible rational zeros of each function. \(f(x)=x^{3}+6 x+2\)

Write a polynomial function of least degree with integral ceefficients the zeros of which include \(\frac{1}{2}, 3,\) and \(-3 .\)

Write each expression in quadratic form, if possible. $$ 84 n^{4}-62 n^{2} $$

STANDARDIZED TEST PRACTICE Which expression is equal to \(\left(x^{2}-4 x+6\right)(x-3)^{-1} ?\) $$ \begin{array}{llll}{\text { A } x-1} & {\text { B } x-1+\frac{3}{x-3}} & {\text { C } x-1-\frac{3}{x-3}} & {\text { D }-x+1-\frac{3}{x-3}}\end{array} $$

Simplify. $$ (2 z-1)(2 z+1) $$

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=x^{3}+2 x^{2}-3 x+1 $$

For Exercises \(11-18,\) complete each of the following. a. Graph each function by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=-x^{3}-4 x^{2} $$

List all of the possible rational zeros of each function. \(h(x)=x^{3}+8 x+6\)

Solve each equation. State the number and type of roots. \(3 x+8=0\)

Solve each equation. $$ x^{4}-50 x^{2}+49=0 $$

Simplify. $$ \left(12 y^{2}+36 y+15\right) \div(6 y+3) $$

Simplify. $$ (2 m-3 n)^{2} $$

Simplify. Assume that no variable equals 0. $$ \left(\frac{1}{3} a^{8} b^{2}\right)\left(2 a^{2} b^{2}\right) $$

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=x^{3}-5 x+2 $$

For Exercises \(11-18,\) complete each of the following. a. Graph each function by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=x^{3}-2 x^{2}+6 $$

List all of the possible rational zeros of each function. \(f(x)=3 x^{4}+15\)

Solve each equation. State the number and type of roots. \(2 x^{2}-5 x+12=0\)

Solve each equation. $$ x^{3}-125=0 $$

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. \(7-x\)

Simplify. $$ (x+1)\left(x^{2}-2 x+3\right) $$

Simplify. $$ \frac{9 b^{2}+9 b-10}{3 b-2} $$

Simplify. Assume that no variable equals 0. $$ \left(5 c d^{2}\right)\left(-c^{4} d\right) $$

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=x^{4}-6 x-8 $$