Chapter 8

Fill in the blanks. A quotient of two polynomials, such as \(\frac{x^{2}+x}{x^{2}-3 x}\), is called a _____ expression.

Fill in the blanks. A set of ordered pairs is called a _____. The set of all first components of the ordered pairs is called the _____ and the set of all second components is called the _____.

The equation \(y=k x\) defines_____ variation: As \(x\) increases,\(y\)_____.

Fill in the blanks. When the polynomial \(4 x^{2}-25\) is written as \((2 x)^{2}-(5)^{2},\) we see that it is the difference of two ________.

Fill in the blanks. When we write \(2 x+4\) as \(2(x+2),\) we say that we have _____ \(2 x+4\).

Fill in the blanks. The _____ ______ of a number is its distance from 0 on a number line.

Fill in the blanks. Functions whose graphs are not lines are called ____ functions.

Fill in the blanks. The ___ of two sets is the set of elements that are common to both sets and the ___of two sets is the set of elements that are in one set, or the other, or both.

Fill in the blanks. An_____ is a statement indicating that two expressions are equal.

Fill in the blanks. A ______ function, such as \(f(x)=\frac{x-7}{x^{2}-x-6},\) is a function whose equation is defined by a rational expression in one variable.

Fill in the blanks. A _____ is a set of ordered pairs (a relation) in which to each first component there corresponds exactly one second component.

Fill in the blanks. When the polynomial \(8 x^{3}+125\) is written as \((2 x)^{3}+(5)^{3},\) we see that it is the sum of two ________.

Fill in the blanks. When we factor a polynomial, we write a sum of terms as a ____ of factors.

Fill in the blanks. \(|2 x-1|=10\) is an absolute value _____ and \(|2 x-1|>10\) is an absolute value _____.

Fill in the blanks. The graph of \(f(x)=x^{2}\) is a cuplike shape called a _____.

Fill in the blanks. \(x \geq 3\) and \(x<4\) is a ___inequality.

Fill in the blanks. \(2 x+1=4\) is an example of a _____ equation in one variable.

Fill in the blanks. The ______ of a function is the set of all permissible input values for the variable.

Fill in the blanks. Given a relation in \(x\) and \(y,\) if to each value of \(x\) in the domain there corresponds exactly one value of \(y\) in the range, \(y\) is said to be a _____ of \(x .\) We call \(x\) the independent _____ and \(y\) the _____ variable.

The equation \(y=k x z\) defines_____ variation, and \(y=\frac{k z}{x}\) defines _____ variation.

a. Write the first ten perfect-square integers. b. Write the first ten perfect-cube integers.

Fill in the blanks. The abbreviation GCF stands for ______ ______ ______.

Fill in the blanks. \(\mathrm{To}\) _______ the absolute value in \(|3-x|-4=5,\) we add 4 to both sides.

Fill in the blanks. The set of _____ real numbers is the set of real numbers greater than or equal to 0.

Fill in the blanks. \(-6

Fill in the blanks. The rational function \(f(x)=\frac{9 x}{x-10}\) is _____ for \(x=10\) In other words, there is a _____ on the domain of the function: \(x \neq 10\).

Fill in the blanks. For a function, the set of all possible values that can be used for the independent variable is called the _____. The set of all values of the dependent variable is called the _____.

The equation \(y=\frac{k x}{z}\) means that \(y\) varies _____ with \(x\) and _____ with \(z\).

a. Use multiplication to verify that the sum of two squares \(x^{2}+25\) does not factor as \((x+5)(x+5)\) b. Use multiplication to verify that the difference of two squares \(x^{2}-25\) factors as \((x+5)(x-5)\)

Fill in the blanks. If a polynomial cannot be factored, it is called _____ polynomial or an irreducible polynomial.

Fill in the blanks. A shift of the graph of a function upward or downward is called a vertical _____.

Fill in the blanks. \((2,8)\) is an example of an open___\(,[-4,0]\) is an example of a___interval, and \((0,9]\) is an example of a half___interval.

Fill in the blanks. If two equations have the same solution set, they are called _____ equations.

Fill in the blanks. To _____ a rational expression, we remove factors common to the numerator and denominator.

Fill in the blanks. A _____ function is a function that can be defined by an equation of the form \(f(x)=m x+b .\) A polynomial function is a function whose equation is defined by a polynomial in _____ variable.

Complete each factorization. a. \(\mathrm{F}^{2}-\mathrm{L}^{2}=(\mathrm{F}+\mathrm{L})(\quad\quad)\) b. \(F^{3}+L^{3}=(F+L)(\quad\quad\quad)\) c. \(\mathrm{F}^{3}-\mathrm{L}^{3}=(\mathrm{F}-\mathrm{L})(\quad\quad\quad)\)

Fill in the blanks. To factor \(a b+6 a+2 b+12\) by ____, we begin by factoring out \(a\) from the first two terms and 2 from the last two terms.

Graph each basic function by plotting points and give its name. a. \(f(x)=x^{2}\) b. \(f(x)=x^{3}\) c. \(f(x)=|x|\)

Fill in the blanks. An equation that is made true by any permissible replacement value for the variable is called an _____ .

Fill in the blanks. We call \(f(x)=x\) the _____ function because it assigns each real number to itself. We call \(f(x)=2\) a _____ function, because for any input \(x,\) the output is always 2.

Factor each binomial. a. \(5 p^{2}+20\) b. \(5 p^{2}-20\) c. \(5 p^{3}+20\) d. \(5 p^{3}+40\)

Fill in the blanks. The trinomial \(4 a^{2}-5 a-6\) is written in_____ powers of \(a\).

Fill in the blanks. The double inequality \(4<3 x+5 \leq 15\) is equivalent to \(4<3 x+5\) ___\(3 x+5 \leq 15\)

Complete each sentence about finding function values graphically. a. To find \(f(-3),\) we find the \(y\) -coordinate of the point on the graph whose \(x\) -coordinate is ___. b. To find the value of \(x\) for which \(f(x)=-2,\) we find the \(x\) -coordinate of the point(s) on the graph whose \(y\) -coordinate is _____. c. Suppose for a function \(f\) that \(f(5)=9 .\) The corresponding ordered pair that will be on the graph of the function is (___,___).

Fill in the blanks. \(f(x)=|6 x-2|\) is called an absolute value _________.

Fill in the blanks. An equation that is false for all replacement values for the variable is called a _____ .

In the rational expression \(\frac{(x+2)(3 x-1)}{(x+2)(4 x+2)},\) the binomial \(x+2\) is a common _____ of the numerator and the denominator.

U.S. Recycling. The following table gives the approximate number of aluminum cans (in billions) collected each year for the years \(2000-2006\). a. Display the data in the table as a relation, that is, as a set of ordered pairs. b. Find the domain and range of the relation. c. Use an arrow diagram to show how members of the range correspond to members of the domain. $$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline \begin{array}{l} \text { Billions of } \\ \text { aluminum cans } \end{array} & 63 & 56 & 54 & 50 & 52 & 51 & 51 \\ \hline \end{array} $$

Tell whether each relationship suggests direct or inverse variation. The amount of money you receive and the number of aluminum cans you return

Give an example of each. a. a difference of two squares b. a square of a difference c. a sum of two squares d. a sum of two cubes e. a cube of a sum