Chapter 6

Graph each polynomial function by making a table of values. $$ f(x)=x^{3}-x^{2}-4 x+4 $$

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

List all of the possible rational zeros of each function. \(p(x)=x^{4}-10\)

Solve each equation. State the number and type of roots. \(x^{2}+4=0\)

Factor completely. If the polynomial is not factorable, write prime. $$ -12 x^{2}-6 x $$

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. \(5 x^{6}-8 x^{2}\)

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

Determine whether each expression is a polynomial. If is a polynomial, state the degree of the polynomial. $$ 2 a+5 b $$

Simplify. Assume that no variable equals 0. $$ \left(-3 x^{2} y^{3}\right)\left(5 x^{5} y^{6}\right) $$

Graph each polynomial function by making a table of values. $$ f(x)=x^{4}-7 x^{2}+x+5 $$

Use synthetic substitution to find \(f(3)\) and \(f(-4)\) for each function. $$ f(x)=5 x^{4}-6 x^{2}+2 $$

List all of the possible rational zeros of each function. \(d(x)=6 x^{3}+6 x^{2}-15 x-2\)

Solve each equation. State the number and type of roots. \(x^{3}+4 x^{2}-21 x=0\)

Factor completely. If the polynomial is not factorable, write prime. $$ a^{2}+5 a+a b $$

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. \(2 b+4 b^{3}-3 b^{5}-7\)

Simplify. \(\left(5 a b^{2}-4 a b+7 a^{2} b\right)(a b)^{-1}\)

Determine whether each expression is a polynomial. If is a polynomial, state the degree of the polynomial. $$ \frac{1}{3} x^{3}-9 y $$

Simplify. Assume that no variable equals 0. $$ \frac{30 y^{4}}{-5 y^{2}} $$

For Exercises \(3-5,\) use the following information. The projected sales of e-books in millions of dollars can be modeled by the function \(S(x)=-17 x^{3}+200 x^{2}-113 x+44,\) where \(x\) is the number of years since 2000 . Use synthetic substitution to estimate the sales for 2008 .

Determine the consecutive integer values of \(x\) between which each real zero of each function is located. Then draw the graph. $$ f(x)=x^{3}-x^{2}+1 $$

Find all of the rational zeros of each function. \(p(x)=x^{3}-5 x^{2}-22 x+56\)

State the possible number of positive real zeros, negative real zeros, and imaginary zeros of each function. \(f(x)=5 x^{3}+8 x^{2}-4 x+3\)

Factor completely. If the polynomial is not factorable, write prime. $$ 21-7 y+3 x-x y $$

Find p(3) and p(-1) for each function. \(p(x)=-x^{3}+x^{2}-x\)

BAKING The number of cookies produced in a factory each day can be estimated by \(C(w)=-w^{2}+16 w+1000\) , where \(w\) is the number of workers and \(C\) is the number of cookies produced. Divide to find the average number of cookies produced per worker.

Determine whether each expression is a polynomial. If is a polynomial, state the degree of the polynomial. $$ \frac{m w^{2}-3}{n z^{3}+1} $$

Simplify. Assume that no variable equals 0. $$ \frac{-2 a^{3} b^{6}}{18 a^{2} b^{2}} $$

For Exercises \(3-5,\) use the following information. The projected sales of e-books in millions of dollars can be modeled by the function \(S(x)=-17 x^{3}+200 x^{2}-113 x+44,\) where \(x\) is the number of years since 2000 . Use direct substitution to evaluate \(S(8)\)

Determine the consecutive integer values of \(x\) between which each real zero of each function is located. Then draw the graph. $$ f(x)=x^{4}-4 x^{2}+2 $$

Find all of the rational zeros of each function. \(f(x)=x^{3}-x^{2}-34 x-56\)

State the possible number of positive real zeros, negative real zeros, and imaginary zeros of each function. \(r(x)=x^{5}-x^{3}-x+1\)

Find p(3) and p(-1) for each function. \(p(x)=x^{4}-3 x^{3}+2 x^{2}-5 x+1\)

Simplify. $$ \left(x^{2}-10 x-24\right) \div(x+2) $$

Simplify. $$ (2 a+3 b)+(8 a-5 b) $$

Simplify. Assume that no variable equals 0. $$ (2 b)^{4} $$

For Exercises \(3-5,\) use the following information. The projected sales of e-books in millions of dollars can be modeled by the function \(S(x)=-17 x^{3}+200 x^{2}-113 x+44,\) where \(x\) is the number of years since 2000 . Which method-synthetic substitution or direct substitution- do you prefer to use to evaluate polynomials? Explain your answer.

Graph each polynomial function. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. State the domain and range for each function. $$ f(x)=x^{3}+2 x^{2}-3 x-5 $$

Find all of the rational zeros of each function. \(t(x)=x^{4}-13 x^{2}+36\)

Find all of the zeros of each function. \(p(x)=x^{3}+2 x^{2}-3 x+20\)

Factor completely. If the polynomial is not factorable, write prime. $$ z^{2}-4 z-12 $$

The intensity of light emitted by a firefly can be determined by \(L(t)=10+0.3 t+0.4 t^{2}-0.01 t^{3},\) where \(t\) is temperature in degrees Celsius and \(L(t)\) is light intensity in lumens. If the temperature is \(30^{\circ} \mathrm{C},\) find the light intensity.

Simplify. $$ \left(3 a^{4}-6 a^{3}-2 a^{2}+a-6\right) \div(a+1) $$

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

Simplify. Assume that no variable equals 0. $$ \left(\frac{1}{w^{4} z^{2}}\right)^{3} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{3}-x^{2}-5 x-3 ; x+1 $$

Graph each polynomial function. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. State the domain and range for each function. $$ f(x)=x^{4}-8 x^{2}+10 $$

Find all of the rational zeros of each function. \(f(x)=2 x^{3}-7 x^{2}-8 x+28\)

Find all of the zeros of each function. \(f(x)=x^{3}-4 x^{2}+6 x-4\)

Factor completely. If the polynomial is not factorable, write prime. $$ 3 b^{2}-48 $$

Simplify. $$ \left(z^{5}-3 z^{2}-20\right) \div(z-2) $$