Intermediate Algebra: Rational Expressions, Equations, and Variation

Rational Expressions and Functions

Introduction to Rational Functions and Equations

A rational function is a function of the form $f(x) = \frac{p(x)}{q(x)}$, where p(x) and q(x) are polynomials, and $q(x) \neq 0$. The domain of a rational function consists of all real numbers except those that make the denominator zero.

• Example: $f(x) = \frac{x^2 + 3}{2x^2 + 4x - 5}$

• Domain: Solve $q(x) = 0$ to find excluded values.

Evaluating Rational Functions: Substitute the given value into the function, ensuring the denominator is not zero.

• Example: $f(1) = \frac{1+1}{1-2} = -2$

• Undefined: $f(2) = \frac{2+1}{2-2}$ is undefined since denominator is zero.

Graphing Rational Functions

Rational functions often have vertical asymptotes where the denominator is zero and horizontal asymptotes determined by the degrees of numerator and denominator.

• As $x$ approaches the value that makes the denominator zero from the left, $f(x)$ may approach $-\infty$; from the right, $+\infty$.

• Vertical Asymptote: $x = a$ if $q(a) = 0$ and $p(a) \neq 0$.

Application Example: For a train curve, $f(r) = \frac{2540}{r}$ gives the elevation needed for the outer rail. As $r$ increases, $f(r)$ decreases, showing an inverse relationship.

Solving Rational Equations

A rational equation contains one or more rational expressions. To solve, multiply both sides by the least common denominator (LCD) to clear fractions, then solve the resulting equation.

• Example: $\frac{3x}{2x-1} = 3$

• Multiply both sides by $2x-1$, solve for $x$.

Applications: Rational equations can model real-world problems, such as average waiting times or combined work rates.

Operations on Functions

Given functions $f$ and $g$, and $x$ in their domains:

• Addition: $(f+g)(x) = f(x) + g(x)$

• Subtraction: $(f-g)(x) = f(x) - g(x)$

• Multiplication: $(fg)(x) = f(x) \cdot g(x)$

• Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

Multiplication and Division of Rational Expressions

Simplifying Rational Expressions

To simplify, factor numerators and denominators, then cancel common factors.

• Example: $\frac{9x}{12x^2} = \frac{3}{4x}$

Multiplying and Dividing Rational Expressions

• Multiply numerators together and denominators together, then simplify.

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

• Example: $\frac{5x}{8} \cdot \frac{4}{10x^2} = \frac{1}{4x}$

Applications

Rational expressions can model average cost, area, and other real-world quantities.

Addition and Subtraction of Rational Expressions

Finding the Least Common Multiple (LCM)

The LCM of two polynomials is the product of each factor the greatest number of times it occurs in either polynomial.

• Example: LCM of $2x$ and $3x$ is $6x$.

Adding and Subtracting Rational Expressions

To add or subtract, first find the LCM of the denominators, rewrite each expression with the LCM, then combine numerators.

• Example: $\frac{1}{2} + \frac{3}{8} = \frac{7}{8}$

Applications

Combined resistance in parallel circuits is found using rational expressions:

• $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$

Solving Rational Equations

Clearing Denominators

Multiply both sides by the LCD to eliminate denominators, then solve the resulting equation.

• Example: $\frac{x}{x+1} = \frac{3}{2(x-1)}$

• Multiply both sides by $2(x+1)(x-1)$, solve for $x$.

Applications

• Work problems: If one pump empties a pool in 50 hours and another in 80 hours, together they take $\frac{400}{13}$ hours.

• Mixture and distance problems can also be modeled with rational equations.

Complex Fractions

Simplifying Complex Fractions

A complex fraction has fractions in its numerator, denominator, or both. Simplify by finding a common denominator or multiplying numerator and denominator by the LCD.

• Example: $\frac{7/8}{3/4} = \frac{7}{6}$

Modeling with Proportions and Variation

Direct Proportion

Quantities $y$ and $x$ are in direct proportion if $y = kx$, where $k$ is the constant of proportionality. The graph is a straight line through the origin.

• Example: If $y = 48$ when $x = 12$, then $k = 4$ and $y = 4x$.

Inverse Variation

Quantities $y$ and $x$ are in inverse proportion if $y = \frac{k}{x}$, where $k$ is a constant. As $x$ increases, $y$ decreases.

• Example: If $y = 16$ when $x = 0.3$, then $k = 4.8$ and $y = \frac{4.8}{x}$.

Joint Variation

When a variable varies directly as the product of two or more variables and/or inversely as another, it is called joint variation.

• Example: $y = k \frac{xz}{w^2}$

Applications

• Stopping distance, volume of trees, and other physical phenomena can be modeled using direct, inverse, or joint variation.

Division of Polynomials

Dividing a Polynomial by a Monomial

Divide each term of the polynomial by the monomial.

• Example: $\frac{5x^3 - 15x^2}{5x^2} = x - 3$

Long Division of Polynomials

Divide the leading term of the dividend by the leading term of the divisor, multiply, subtract, and repeat until the degree of the remainder is less than the divisor.

• Example: $\frac{2x^3 - 5x^2 + 4x - 1}{x - 1}$ yields quotient $2x^2 - 3x + 1$.

References: Intermediate Algebra with Applications and Visualizations, Rockswold/Krieger/Rockswold, Pearson.