Chapter 6

Fill in the blanks. The method of dividing \(x^{2}+2 x-9\) by \(x-4\) shown below is called ____ division. $$ \begin{array}{rrr} 1 & 2 & -9 \\ & 4 & 24 \\ \hline 1 & 6 & 15 \end{array} $$

Fill in the blanks. \(\frac{\frac{x}{y}+\frac{1}{x}}{\frac{1}{y}+\frac{2}{x}}\) and \(\frac{\frac{5 a^{2}}{b}}{\frac{b}{2 a^{3}}}\) are examples of complex _____ expressions, or more simply, _____ fractions.

A _________ is the quotient of two numbers or two quantities with the same units.

Fill in the blanks. Equations that contain one or more rational expressions, such as \(\frac{x}{x+2}=4+\frac{10}{x+1},\) are called ______ equations.

Fill in the blanks. The rational expressions \(\frac{7}{60}\) and \(\frac{n+1}{6 n}\) have a common _____ of \(6 n\).

Fill in the blanks. The expression \(\frac{18 x^{7}}{9 x^{4}}\) is a monomial divided by a _________. The expression \(\frac{6 x^{3}-4 x^{2}+8 x-2}{2 x^{4}}\) is a _________ divided by a monomial. The expression \(\frac{x^{2}-8 x+12}{x-6}\) is a trinomial divided by a _________.

Fill in the blanks. $$ \frac{a^{2}-9}{a^{2}-49} \cdot \frac{a-7}{a+3} \text { is the product of two_____expressions. } $$

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. Synthetic division is used to divide a polynomial by a ____ of the form \(x-k\)

Fill in the blanks. To _____ a complex fraction means to express it in the form \(\frac{A}{B},\) where \(A\) and \(B\) are polynomials with no common factors.

An equation that states that two ratios are equal, such as \(\frac{1}{2}=\frac{4}{8}\) is called a ________.

Fill in the blanks. When a boat travels ____ , the speed of the boat is increased by the current. When a boat travels____, the speed of the boat is decreased by the current.

Fill in the blanks. When solving a rational equation, if we obtain a number that does not satisfy the original equation, the number is called an ___________ solution.

Fill in the blanks. The powers of \(x\) in \(2 x^{4}+3 x^{3}+4 x^{2}-7 x-8\) are written in _________ order.

Fill in the blanks. The _____ of \(\frac{a+3}{a+7}\) is \(\frac{a+7}{a+3}\)

To simplify the following complex fraction, it is multiplied by what form of 1? $$ \frac{\frac{4}{t^{2}}+\frac{b}{t}}{\frac{3 b}{t}}=\frac{\frac{4}{t^{2}}+\frac{b}{t}}{\frac{3 b}{t}} \cdot \frac{t^{2}}{t^{2}} $$

In \(\frac{50}{3}=\frac{x}{9},\) the terms 50 and 9 are called the _________ and the terms 3 and \(x\) are called the _________ of the proportion. In a the product of the _________ is equal to the product of the _________.

Fill in the blank: If a job can be completed in \(x\) hours, then the rate of work can be expressed as \(\frac{1}{\underline{\phantom{xx}}}\) of the job is completed per hour.

Classify each of the following as an expression or an equation. a. \(\frac{7}{5 x}-\frac{1}{2}=\frac{5}{6 x}+\frac{1}{3}\) b. \(\frac{4}{x^{2}-4}-\frac{5}{x-2}\) c. \(\frac{27 p^{4}}{35 q} \div \frac{9 p}{21 q}\) d. \(\frac{4}{t+3}+\frac{8}{t^{2}-9}=\frac{2}{t-3}\) e. \(\frac{\frac{y}{x}-\frac{x}{y}}{\frac{1}{y}-\frac{1}{x}}\) f. \(\frac{t^{2}+t-6}{t^{2}-6 t+9} \cdot \frac{t^{2}-9}{t^{2} 4}\)

Fill in the blanks. To ______ a rational expression, we multiply it by a form of 1 For example, \(\frac{2}{n^{2}} \cdot \frac{8}{8}=\frac{16}{8 n^{2}}\).

Fill in the blanks. To find the reciprocal of a rational expression, we ____ its numerator and denominator.

Fill in the blanks. 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.

Fill in the blanks. By the ____ theorem, if a polynomial \(P(x)\) is divided by \(x-k,\) the remainder is \(P(k)\).

Determine the LCD of the rational expressions appearing in each complex fraction. a \(\frac{1+\frac{4}{c}}{\frac{2}{c}+c} \quad\) b. \(\frac{\frac{6}{m^{2}}+\frac{1}{2 m}}{\frac{m^{2}-1}{4}}\) c. \(\frac{\frac{p}{p+2}+\frac{12}{p+3}}{\frac{p-1}{p^{2}+5 p+6}}\) d. \(\frac{2+\frac{3}{x+1}}{\frac{1}{x}+x+x^{2}}\)

The ________ products for the proportion \(\frac{10}{3}=\frac{5}{x}\) are \(10 x\) and 15

Check to determine whether 2 is a solution of the following equations. a. \(\frac{x}{2}+\frac{4}{x+2}=x\) b. \(\frac{x+2}{x-2}+\frac{1}{x^{2}-4}=1\)

Fill in the blanks. The polynomials \(x-y\) and \(y-x\) are ______ because their terms are the same but opposite in sign.

since \(5 x^{2}+6\) is missing an \(x\) -term, we insert a _________ \(0 x\) term in a division and write the polynomial as \(5 x^{2}+0 x+6\)

Fill in the blanks. To simplify a rational expression, remove any factors ____ to the numerator and denominator.

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

Fill in the blanks. Method 1: To simplify a complex fraction, write its numerator and denominator as _____ rational expressions. Then perform the indicated _____ by multiplying the numerator of the complex fraction by the _____ of the denominator.

If two angles of one triangle have the same measure as two angles of a second triangle, the triangles are _________.

Consider the rational equation: \(\frac{x}{x-3}=\frac{1}{x}+\frac{2}{x-3}\). a. What values of \(x\) make a denominator \(0 ?\) b. What values of \(x\) make a rational expression undefined? c. What numbers can't be solutions of the equation?

Fill in the blanks. To add or subtract rational expressions that have the same denominator, add or subtract the _____ and write the sum or difference over the common _____ In symbols, if \(\frac{A}{D}\) and \(\frac{B}{D}\) are rational expressions, \(\frac{A}{D}+\frac{B}{D}=\frac{\underline{\phantom{xx}}}{D}\) and \(\frac{A}{D}-\frac{B}{D}=\frac{\underline{\phantom{xx}}}{D}\)

Fill in the blanks. a. To divide a polynomial by a monomial, divide each _________ of the polynomial by the monomial. b. \(\frac{18 x+9}{9}=\frac{18 x}{\underline{\phantom{xx}}}+\frac{9}{\underline{\phantom{xx}}}\) c. \(\frac{30 x^{2}+12 x-24}{6}=\frac{30 x^{2}}{\underline{\phantom{xx}}}+\frac{12 x}{\underline{\phantom{xx}}}-\frac{24}{\underline{\phantom{xx}}}\)

Fill in the blanks. Because of the division by \(0,\) the expression \(\frac{8}{0}\) is ___.

Fill in the blanks. If \(P(x)\) is a polynomial and if \(P(k)=0,\) then \(k\) is called a ____ of the polynomial.

Fill in the blanks. Method 2: To simplify a complex fraction, find the LCD of _____ rational expressions within the complex fraction. Multiply the complex fraction by 1 in the form \(\square\).

The equation \(y=k x\) defines ___________ variation: As \(x\) increases, y ________.

Solve \(d=r t\) for \(t\)

To clear the following equations of fractions, by what should both sides be multiplied? a. \(\frac{1}{a}=\frac{1}{3}-\frac{2}{3 a}\) b. \(\frac{2}{x-2}+\frac{10}{x+5}=\frac{2 x}{x^{2}+3 x-10}\)

Fill in the blanks. To find the least common denominator of several rational expressions, _____ each denominator completely. The LCD is a product that uses each different factor the _____ number of times it appears in any one factorization.

Fill in the blanks. The binomials \(x-15\) and \(15-x\) are called ___ because their terms are the same, except that they are opposite in sign.

Suppose that after dividing \(2 x^{3}+5 x^{2}-11 x+4\) by \(2 x-1\) you obtain \(x^{2}+3 x-4\). Show how multiplication can be used to check the result.

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

Complete each solution to simplify the rational expression. $$ \frac{\frac{2}{a}-\frac{1}{b}}{\frac{5}{a}+\frac{3}{b}}=\frac{\frac{2}{a}-\frac{1}{b}}{\frac{5}{a}+\frac{3}{b}} \cdot \frac{\square}{\square} $$ $$ =\frac{\left(\frac{2}{a}-\frac{1}{b}\right)}{\left(\frac{5}{a}+\frac{3}{b}\right)} \frac{\square}{\square} $$ $$ =\frac{\frac{2}{a} \cdot \square}{\frac{5}{a} \cdot \square}+\frac{\frac{1}{b} \cdot \square}{b \cdot \square} $$ $$ =\frac{2 b- \square}{\square +3 a} $$

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

Perform each multiplication. a. \(4 x\left(\frac{3}{4 x}\right)\) b. \((x+6)(x-2)\left(\frac{3}{x-2}\right)\) c. \(8(x+4)\left[\frac{7 x}{2(x+4)}\right]\) d. \(6(m-5)\left(\frac{7}{5-m}\right)\)

A boat can cruise at 30 mph in still water. a. What is its cruising speed upstream against a current of \(4 \mathrm{mph} ?\) b. What is its cruising speed downstream with a current of \(4 \mathrm{mph} ?\)

$$\frac{x+3}{x+3}=\square$$