Chapter 6

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. $$ f(x)=x^{3}-3 x^{2}+2 $$

List all of the possible rational zeros of each function. \(n(x)=x^{5}+6 x^{3}-12 x+18\)

Solve each equation. State the number and type of roots. \(x^{3}+9 x=0\)

POOL. The Shelby University swimming pool is in the shape of a rectangular prism and has a volume of \(28,000\) cubic feet. The dimensions of the pool are \(x\) feet deep by \(7 x-6\) feet wide by \(9 x-2\) feet long. How deep is the pool?

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. \((a+1)\left(a^{2}-4\right)\)

Simplify. $$ \frac{9 a^{3} b^{2}-18 a^{2} b^{3}}{3 a^{2} b} $$

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

Simplify. Assume that no variable equals 0. $$ \left(7 x^{3} y^{-5}\right)\left(4 x y^{3}\right) $$

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=2 x^{3}-8 x^{2}-2 x+5 $$

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. $$ f(x)=x^{3}+5 x^{2}-9 $$

List all of the possible rational zeros of each function. \(p(x)=3 x^{3}-5 x^{2}-11 x+3\)

Solve each equation. State the number and type of roots. \(x^{4}-81=0\)

Factor completely. If the polynomial is not factorable, write prime. $$ 2 x y^{3}-10 x $$

Simplify. $$ \frac{5 x y^{2}-6 y^{3}+3 x^{2} y^{3}}{x y} $$

Simplify. Assume that no variable equals 0. $$ \left(-3 b^{3} c\right)\left(7 b^{2} c^{2}\right) $$

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=3 x^{4}+x^{3}-2 x^{2}+x+12 $$

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. $$ f(x)=-3 x^{3}+20 x^{2}-36 x+16 $$

List all of the possible rational zeros of each function. \(h(x)=9 x^{6}-5 x^{3}+27\)

Solve each equation. State the number and type of roots. \(x^{4}-16=0\)

Factor completely. If the polynomial is not factorable, write prime. $$ 6 a^{2} b^{2}+18 a b^{3} $$

Simplify. $$ \left(28 c^{3} d-42 c d^{2}+56 c d^{3}\right) \div(14 c d) $$

Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. $$ 3 z^{2}-5 z+11 $$

Simplify. Assume that no variable equals 0. $$ \frac{a^{2} n^{6}}{a n^{5}} $$

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=x^{5}+8 x^{3}+2 x-15 $$

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. $$ f(x)=x^{3}-4 x^{2}+2 x-1 $$

Find all of the rational zeros of each function. \(f(x)=x^{3}+x^{2}-80 x-300\)

Solve each equation. State the number and type of roots. \(x^{5}-8 x^{3}+16 x=0\)

Factor completely. If the polynomial is not factorable, write prime. $$ 12 c d^{3}-8 c^{2} d^{2}+10 c^{5} d^{3} $$

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. \(6 x^{4}+3 x^{2}+4 x-8\)

Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. $$ x^{3}-9 $$

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

Simplify. Assume that no variable equals 0. $$ \frac{-y^{5} z^{7}}{y^{2} z^{5}} $$

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=x^{6}-4 x^{4}+3 x^{2}-10 $$

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. $$ f(x)=x^{4}-8 $$

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

State the number of positive real zeros, negative real zeros, and imaginary zeros for each function. \(f(x)=x^{3}-6 x^{2}+1\)

Factor completely. If the polynomial is not factorable, write prime. $$ 3 a^{2} b x+15 c x^{2} y+25 a d^{3} y $$

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. \(7+3 x^{2}-5 x^{3}+6 x^{2}-2 x\)

Simplify. $$ \left(x^{3}-4 x^{2}\right) \div(x-4) $$

Simplify. Assume that no variable equals 0. $$ \frac{-5 x^{3} y^{3} z^{4}}{20 x^{3} y^{7} z^{4}} $$

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

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. $$ f(x)=x^{4}-10 x^{2}+9 $$

Find all of the rational zeros of each function. \(f(x)=2 x^{5}-x^{4}-2 x+1\)

Factor completely. If the polynomial is not factorable, write prime. $$ 8 y z-6 z-12 y+9 $$

Find \(p(4)\) and \(p(-2)\) for each function. \(p(x)=2-x\)

Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. $$ \sqrt{m-5} $$

Simplify. $$ \left(x^{3}-27\right) \div(x-3) $$

Simplify. Assume that no variable equals 0. $$ \frac{3 a^{5} b^{3} c^{3}}{9 a^{3} b^{7} c} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{3}-x^{2}-10 x-8 ; x+1 $$

Find all of the rational zeros of each function. \(f(x)=x^{5}-6 x^{3}+8 x\)