Chapter 6

Simplify. $$ d^{-3}\left(d^{5}-2 d^{3}+d^{-1}\right) $$

ACT/SAT What is the remainder when \(x^{3}-7 x+5\) is divided by \(x+3 ?\) \(\mathbf{A}-11\) \(\mathbf{B}-1\) \(\mathbf{C} 1\) \(\mathbf{D} 11\)

REVIEW. The total area of a rectangle is \(25 a^{4}-16 b^{2} .\) Which factors could represent the length times width? \(\mathbf{F}\left(5 a^{2}+4 b\right)\left(5 a^{2}+4 b\right)\) \(\mathbf{G}\left(5 a^{2}+4 b\right)\left(5 a^{2}-4 b\right)\) \(\mathbf{H}(5 a-4 b)(5 a-4 b)\) \(\mathbf{J}(5 a+4 b)(5 a-4 b)\)

Given a function and one of its zeros, find all of the zeros of the function. \(h(x)=x^{3}-11 x+20 ; 2+i\)

Find the number of regions formed by connecting 5 points of a circle. Draw a diagram to verify your solution.

Factor completely. If the polynomial is not factorable, write prime. $$ 3 a^{3}+2 a^{2}-5 a+9 a^{2} b+6 a b-15 b $$

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

REVIEW If \(i=\sqrt{-1},\) then \(5 i(7 i)=\) \(\mathrm{F} 70\) \(\mathrm{G} 35\) \(\mathrm{H}-35\) \(\mathrm{J}-70\)

ACT/SAT Which expression is equal to \(\frac{\left(2 x^{2}\right)^{3}}{12 x^{4}} ?\) $$ \begin{array}{ll}{\mathbf{A} \frac{x}{2}} & {\mathbf{C} \frac{1}{2 x^{2}}} \\\ {\mathbf{B} \frac{2 x}{3}} & {\mathbf{D} \frac{2 x^{2}}{3}}\end{array} $$

Factor completely. If the polynomial is not factorable, write prime. $$ 7 x y^{3}-14 x^{2} y^{5}+28 x^{3} y^{2} $$

If \(p(x)=2 x^{2}-5 x+4\) and \(r(x)=3 x^{3}-x^{2}-2,\) find each value. $$ p\left(2 a^{2}\right) $$

Given a function and one of its zeros, find all of the zeros of the function. \(f(x)=x^{3}+5 x^{2}+9 x+45 ;-5\)

State the least degree a polynomial equation with real coefficients can have if it has roots at \(x=5+i, x=3-2 i,\) and a double root at \(x=0 .\) Explain.

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

Simplify. $$ \left(2 x^{2}-3 x+5\right)-\left(3 x^{2}+x-9\right) $$

REVIEW. Four students worked the same math problem. Each student's work is shown below. $$ \begin{array}{ll}{\frac{\text { Student } \mathrm{F}}{x^{2} x^{-5}=\frac{x^{2}}{x^{5}}}} & {\frac{\text { Student } \mathrm{G}}{x^{2} x^{-5}=\frac{x^{2}}{x^{-5}}}} \\ {=\frac{1}{x^{3}}, x \neq 0} & {=x^{7}, x \neq 0}\end{array} $$ $$ \begin{array}{l}{\frac{\text { Student } \mathrm{H}}{x^{2} x^{-5}=\frac{x^{2}}{x^{-5}}}} \\ {\quad=x^{-7}, x \neq 0}\end{array} $$ $$ \begin{array}{l}{\text { Student I }} \\ {\begin{aligned} x^{2} x^{-5} &=\frac{x^{2}}{x^{5}} \\ &=x^{3}, x \neq 0 \end{aligned}}\end{array} $$ Which is a completely correct solution? $$ \begin{array}{ll}{\mathbf{F} \text { Student } \mathrm{F}} & {\mathbf{H} \text { Student } \mathrm{H}} \\ {\text { G Student } \mathrm{G}} & {\text { J Student } \mathrm{J}}\end{array} $$

Factor completely. If the polynomial is not factorable, write prime. $$ a b-5 a+3 b-15 $$

Given a function and one of its zeros, find all of the zeros of the function. \(g(x)=x^{3}-3 x^{2}-41 x+203 ;-7\)

If \(p(x)=2 x^{2}-5 x+4\) and \(r(x)=3 x^{3}-x^{2}-2,\) find each value. $$ r(x-1) $$

Find a counterexample to disprove the following statement. The polynomial function of least degree with integral coefficients with zeros at \(x=4, x=-1,\) and \(x=3,\) is unique.

Explain why a constant polynomial such as \(f(x)=4\) has degree 0 and a linear polynomial such as \(f(x)=x+5\) has degree 1

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

Simplify. $$ y^{2} z\left(y^{2} z^{3}-y z^{2}+3\right) $$

Solve each inequality algebraically. $$ x^{2}-8 x+12<0 $$

Factor completely. If the polynomial is not factorable, write prime. $$ 2 x^{2}+15 x+25 $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. \(20 x^{3}-29 x^{2}-25 x+6 ; x-2\)

If \(p(x)=2 x^{2}-5 x+4\) and \(r(x)=3 x^{3}-x^{2}-2,\) find each value. $$ p\left(x^{2}+4\right) $$

Sketch the graph of an odd-degree polynomial function with a negative leading coefficient and three real roots.

LANDSCAPING. A boardwalk that is \(x\) feet wide is built around a rectangular pond. The pond is 30 feet wide and 40 feet long. The combined area of the pond and the boardwalk is 2000 square feet. What is the width of the boardwalk?

Simplify. $$ (x-3 y)^{2} $$

Simplify. $$ (y+5)(y-3) $$

Solve each inequality algebraically. $$ x^{2}+2 x-86 \geq-23 $$

Factor completely. If the polynomial is not factorable, write prime. $$ c^{3}-216 $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. \(3 x^{4}-21 x^{3}+38 x^{2}-14 x+24 ; x-3\)

If \(p(x)=2 x^{2}-5 x+4\) and \(r(x)=3 x^{3}-x^{2}-2,\) find each value. $$ 2\left[p\left(x^{2}+1\right)\right]-3 r(x-1) $$

How many negative real zeros does \(f(x)=x^{5}-2 x^{4}-4 x^{3}+\) \(4 x^{2}-5 x+6\) have? A. 3 B. 2 C. 1 D. 0

CHECK FACTORING. Use a graphing calculator to determine if each polynomial is factored correctly. Write yes or no. If the polynomial is not factored correctly, find the correct factorization. $$ 3 x^{2}+5 x+2 \stackrel{?}{=}(3 x+2)(x+1) $$

Simplify. $$ (1+4 c)^{2} $$

Simplify. $$ (a-b)^{2} $$

Solve each inequality algebraically. $$ 15 x^{2}+4 x+12 \leq 0 $$

Graph each function by making a table of values. $$ f(x)=x^{3}-4 x^{2}+x+5 $$

The perimeter of a right triangle is 24 centimeters. Three times the length of the longer leg minus two times the length of the shorter leg exceeds the hypotenuse by 2 centimeters. What are the lengths of all three sides?

Simplify. $$ \left(4 x^{3}-7 x^{2}+3 x-2\right) \div(x-2) $$

Tiles numbered from 1 to 6 are placed in a bag and are drawn out to determine which of six tasks will be assigned to six people. What is the probability that the tiles numbered 5 and 6 are drawn consecutively? F. \(\frac{2}{3}\) G. \(\frac{2}{5}\) H. \(\frac{1}{2}\) J. \(\frac{1}{3}\)

The graph of the polynomial function \(f(x)=a x(x-4)(x+1)\) goes through thepoint at \((5,15) .\) Find the value of \(a\)

CHECK FACTORING. Use a graphing calculator to determine if each polynomial is factored correctly. Write yes or no. If the polynomial is not factored correctly, find the correct factorization. $$ x^{3}+8 \stackrel{?}{=}(x+2)\left(x^{2}-x+4\right) $$

GENETICS. Suppose \(R\) and \(W\) represent two genes that a plant can inherit from its parents. The terms of the expansion of \((R+W)^{2}\) represent the possible pairings of the genes in the offspring. Write \((R+W)^{2}\) as a polynomial.

ASTRONOMY Earth is an average of \(1.5 \times 10^{11}\) meters from the Sun. Light travels at \(3 \times 10^{8}\) meters per second. About how long does it take sunlight to reach Earth?

Graph each function. $$ y=-2(x-2)^{2}+3 $$

Graph each function by making a table of values. $$ f(x)=x^{4}-6 x^{3}+10 x^{2}-x-3 $$