Rational Expressions and Functions: Operations, Applications, and Variation

Section 6.1: Rational Functions and Multiplying and Dividing Rational Expressions

Rational Expressions and Functions

A rational expression is a fraction in which both the numerator and denominator are polynomials, with the denominator not equal to zero. A rational function is a function defined by a rational expression.

• General Form: $\frac{P}{Q}$, where $P$ and $Q$ are polynomials and $Q \neq 0$.

• Function Form: $f(x) = \frac{P(x)}{Q(x)}$

Finding the Domain of a Rational Expression

• The domain is all real numbers except those that make the denominator zero.

• Example: $f(x) = \frac{3x}{x-7}$ has domain $x \neq 7$.

• Example: $C(x) = \frac{x^2 + 3}{4 - x}$ has domain $x \neq 4$.

Simplifying Rational Expressions

• Fundamental Principle: For any nonzero polynomial $R$, $\frac{P}{Q} = \frac{PR}{QR}$.

• Steps:

  1. Completely factor the numerator and denominator.

  2. Divide out common factors (removing a factor of 1).

• Example: $\frac{x^2 - 8x}{16 - 8x}$ Factor numerator: $x(x-8)$ Factor denominator: $8(2 - x)$ Additional info: To simplify, factor out negatives if needed and cancel common factors.

Multiplying Rational Expressions

• Rule: $\frac{P}{Q} \cdot \frac{R}{S} = \frac{PR}{QS}$, $Q \neq 0$, $S \neq 0$.

• Steps:

  1. Completely factor each numerator and denominator.

  2. Multiply numerators and denominators.

  3. Simplify by dividing by common factors.

Dividing Rational Expressions

• Rule: $\frac{P}{Q} \div \frac{R}{S} = \frac{P}{Q} \cdot \frac{S}{R} = \frac{PS}{QR}$, $Q, S, R \neq 0$.

• To divide, multiply by the reciprocal of the divisor and simplify.

Applications of Rational Functions

• Example: Total revenue from book sales: $R(x) = \frac{1000x^2}{x^2 + 4}$, where $x$ is years since publication.

• Find $R(1)$ and $R(2)$ for revenue at the end of the first and second years.

• Revenue during the second year: $R(2) - R(1)$.

• Domain: All real $x$ such that $x^2 + 4 \neq 0$ (which is all real numbers, since $x^2 + 4 > 0$ for all real $x$).

Section 6.2: Adding and Subtracting Rational Expressions

Adding/Subtracting with Common Denominators

To add or subtract rational expressions with the same denominator, add or subtract the numerators and keep the common denominator.

• Formulas:

  • $\frac{P}{Q} + \frac{R}{Q} = \frac{P + R}{Q}$

  • $\frac{P}{Q} - \frac{R}{Q} = \frac{P - R}{Q}$

• Example: $\frac{5}{x-2} - \frac{5}{x-2} = 0$

Finding the Least Common Denominator (LCD)

• Factor each denominator completely.

• The LCD is the product of all unique factors, each raised to the highest power that appears in any denominator.

• Example: For $\frac{1}{a^2 - b^2}$ and $\frac{2}{a^2b + ab^2}$, factor denominators and find LCD.

Adding/Subtracting with Unlike Denominators

• Steps:

  1. Find the LCD of the rational expressions.

  2. Rewrite each expression with the LCD as the denominator.

  3. Add or subtract numerators and write the result over the LCD.

  4. Simplify the resulting rational expression.

• Example: $\frac{4}{3x} + \frac{3}{2x}$ LCD is $6x$; rewrite and add: $\frac{8}{6x} + \frac{9}{6x} = \frac{17}{6x}$

Section 6.3: Simplifying Complex Fractions

Complex Fractions

A complex fraction is a fraction whose numerator, denominator, or both contain rational expressions.

Method 1: Simplify Numerator and Denominator, then Divide

• Simplify the numerator and denominator to single fractions.

• Divide by multiplying the numerator by the reciprocal of the denominator.

• Simplify if possible.

• Example: $\frac{\frac{10}{3x}}{\frac{5}{6x}} = \frac{10}{3x} \cdot \frac{6x}{5} = \frac{60x}{15x} = 4$

Method 2: Multiply by the LCD

• Multiply numerator and denominator by the LCD of all fractions in both numerator and denominator.

• Simplify the resulting expression.

Simplifying Expressions with Negative Exponents

• Rewrite expressions so that all exponents are positive.

• Example: $a^{-1}b^{-2} = \frac{1}{ab^2}$

Section 6.4: Dividing Polynomials: Long Division and Synthetic Division

Dividing by a Monomial

• Divide each term in the polynomial by the monomial.

• Example: $\frac{4x^2y + 6xy^2 - 2y^2}{2xy}$ $= \frac{4x^2y}{2xy} + \frac{6xy^2}{2xy} - \frac{2y^2}{2xy} = 2x + 3y - \frac{y}{x}$

Dividing by a Polynomial (Long Division)

• Use long division when dividing by a polynomial with two or more terms.

• Example: Divide $5x^3 - 4x^2 + 12x - 8$ by $x - 2$ using long division.

Synthetic Division

• Used when dividing by a binomial of the form $x - c$.

• If a power of $x$ is missing, insert a zero as its coefficient.

• The last number in the synthetic division process is the remainder.

• The degree of the quotient is one less than the degree of the dividend.

Remainder Theorem

• If a polynomial $P(x)$ is divided by $x - c$, the remainder is $P(c)$.

• Example: For $P(x) = 4x^4 + 2x - 1$, the remainder when divided by $x + 1$ is $P(-1)$.

Section 6.5: Solving Equations Containing Rational Expressions

Solving Rational Equations

• An equation contains an equals sign; an expression does not.

• Both sides of an equation can be multiplied by any nonzero number.

• Steps:

  1. Multiply both sides by the LCD of all rational expressions in the equation.

  2. Simplify both sides.

  3. Solve the resulting equation (linear, quadratic, etc.).

  4. Check the solution in the original equation (to avoid extraneous solutions).

• Example: Solve $\frac{2}{x} + 1 = \frac{5}{2}$ Multiply both sides by $2x$ (LCD), solve for $x$.

Section 6.6: Rational Equations and Problem Solving

Solving for a Specified Variable

• Isolate the specified variable using algebraic operations, including multiplying both sides by the LCD if necessary.

• Example: Solve $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$ for $R$.

Number Problems with Rational Equations

• Translate word problems into equations containing rational expressions.

• Example: The sum of a number and 5 times its reciprocal is 6: $x + \frac{5}{x} = 6$.

Proportion Problems

• A proportion is an equation stating that two ratios are equal.

• Example: If a camel drinks 15 gallons in 10 minutes, how much in 3 minutes? Set up $\frac{15}{10} = \frac{x}{3}$ and solve for $x$.

Work Problems

• When multiple workers complete a job together, their combined rate is the sum of their individual rates.

• Table:

Worker	Minutes to Complete Job	Part of Job per Minute
1	20	$\frac{1}{20}$
2	30	$\frac{1}{30}$
3	60	$\frac{1}{60}$
Together	$t$	$\frac{1}{t}$

• Set up equation: $\frac{1}{20} + \frac{1}{30} + \frac{1}{60} = \frac{1}{t}$

Distance, Rate, and Time Problems

• Use the formula $d = rt$ (distance = rate × time).

• Example: If a boat travels 20 miles downstream and 10 miles upstream in the same time, with current speed 5 mph, find the boat's speed in still water.

• Let $x$ = speed in still water. Downstream rate: $x + 5$, upstream rate: $x - 5$.

• Set up equation: $\frac{20}{x+5} = \frac{10}{x-5}$

Section 6.7: Variation and Problem Solving

Direct Variation

• Definition: $y$ varies directly as $x$ if $y = kx$, where $k$ is the constant of variation.

• Example: If $y = 4$ when $x = 20$, then $k = \frac{y}{x} = \frac{4}{20} = 0.2$; so $y = 0.2x$.

• Application: Weight of a ball varies directly with the cube of its radius: $W = kr^3$.

Inverse Variation

• Definition: $y$ varies inversely as $x$ if $y = \frac{k}{x}$.

• Example: If $y = 6$ when $x = 5$, then $k = yx = 30$; so $y = \frac{30}{x}$.

• Application: Speed $R$ varies inversely with time $T$: $R = \frac{k}{T}$.

Joint Variation

• Definition: $y$ varies jointly as $x$ and $z$ if $y = kxz$.

• For higher powers: $y = kx z^3$ (e.g., varies jointly as $x$ and the cube of $z$).

• Example: If $y = 120$ when $x = 5$ and $z = 2$, then $k = \frac{y}{x z^3} = \frac{120}{5 \times 8} = 3$; so $y = 3x z^3$.

Combined Variation

• Combined variation involves both direct (joint) and inverse variation.

• Example: The maximum weight $W$ a beam can support varies jointly as its width $w$ and the square of its height $h$, and inversely as its length $l$: $W = k \frac{w h^2}{l}$

• Given values, solve for $k$ and use the equation to find $W$ for other dimensions.