Identifying Rational Expressions

Which of these are rational expressions? a) $\frac{x^2 + 3x}{x - 1}$ b) $\frac{\sqrt{x}}{x + 2}$ c) $\frac{5}{x^3 - 8}$

Check expression (a): Numerator: $x^2 + 3x$ (polynomial). Denominator: $x - 1$ (polynomial) = YES - rational expression

Check expression (b): Numerator: $\sqrt{x}$ (NOT a polynomial - has a radical) = NO - not a rational expression

Check expression (c): Numerator: $5$ (constant polynomial). Denominator: $x^3 - 8$ (polynomial) = YES - rational expression

Answer: Expressions (a) and (c) are rational expressions. Expression (b) is not because $\sqrt{x}$ is not a polynomial.

Finding Domain Restrictions

Find the domain restrictions for $\frac{x + 5}{x^2 - 9}$

Set denominator equal to zero: $x^2 - 9 = 0$ = Equation to solve

Factor the denominator: $x^2 - 9 = (x + 3)(x - 3)$ = Difference of squares

Solve for x: $(x + 3)(x - 3) = 0$ $x = -3$ or $x = 3$ = Excluded values found

State the restrictions: The expression is undefined when $x = -3$ or $x = 3$ = $x \neq -3, x \neq 3$

Answer: Domain restrictions: $x \neq -3$ and $x \neq 3$. The expression is defined for all real numbers except $-3$ and $3$.

Evaluating Rational Expressions

Evaluate $\frac{x^2 - 4}{x + 2}$ when $x = 5$

Check for restrictions first: Denominator: $x + 2 = 0$ when $x = -2$ Since $5 \neq -2$, we can proceed = No restriction issue

Substitute $x = 5$ in numerator: $5^2 - 4 = 25 - 4 = 21$ = Numerator = $21$

Substitute $x = 5$ in denominator: $5 + 2 = 7$ = Denominator = $7$

Divide: $\frac{21}{7} = 3$ = Final answer

Answer: When $x = 5$, the expression equals $3$

Multiple Restrictions from Factoring

Find all values that make $\frac{2x + 1}{x^2 - 5x + 6}$ undefined

Set denominator equal to zero: $x^2 - 5x + 6 = 0$ = Equation to solve

Factor the quadratic: Find two numbers that multiply to $6$ and add to $-5$ Those are $-2$ and $-3$ = $(x - 2)(x - 3) = 0$

Solve each factor: $x - 2 = 0 \Rightarrow x = 2$ $x - 3 = 0 \Rightarrow x = 3$ = Two excluded values

State the answer: The expression is undefined at $x = 2$ and $x = 3$ = $x \neq 2, x \neq 3$

Answer: The expression is undefined when $x = 2$ or $x = 3$

Mistake: Setting the numerator equal to zero instead of the denominator

Why: When finding restrictions, we need to find when the expression is undefined. Division by zero is undefined, not division into zero.

Correct: Always set the DENOMINATOR equal to zero to find restrictions. When the numerator is zero, the expression simply equals zero.

Mistake: Forgetting to factor the denominator completely

Why: A quadratic denominator like $x^2 - 9$ has TWO roots, so there are TWO restrictions.

Correct: Always factor the denominator completely before solving. $x^2 - 9 = (x + 3)(x - 3)$ gives restrictions at $x = -3$ AND $x = 3$.

Mistake: Thinking $\frac{0}{5}$ and $\frac{5}{0}$ are the same

Why: $\frac{0}{5} = 0$ (zero divided by anything is zero), but $\frac{5}{0}$ is undefined (cannot divide by zero).

Correct: Zero in the numerator gives zero. Zero in the denominator is undefined.

Average Speed Problems

When traveling different distances at different speeds, the average speed is a rational expression.

If you drive 100 km at speed $v$ and return at speed $v + 20$, the average speed for the whole trip is $\frac{2 \times 100}{\frac{100}{v} + \frac{100}{v+20}}$

Work Rate Problems

When two people work together, their combined rate involves rational expressions.

If Alice completes a job in $x$ hours and Bob in $y$ hours, working together they complete it in $\frac{xy}{x + y}$ hours.

Electrical Circuits

Parallel resistance formulas use rational expressions.

For two resistors in parallel: $R_{total} = \frac{R_1 \times R_2}{R_1 + R_2}$

Q: What makes an expression 'rational'?

A: The term 'rational' comes from 'ratio.' A rational expression is a ratio (fraction) of two polynomials, just like a rational number is a ratio of two integers.

Q: Why can't we divide by zero?

A: Division asks 'how many times does the divisor fit into the dividend?' Zero fits into any number infinitely many times, making the answer undefined. There's no number that, when multiplied by zero, gives a non-zero result.

Q: Is $\frac{5}{x}$ a rational expression?

A: Yes! The number $5$ is a constant polynomial (degree 0), and $x$ is a polynomial (degree 1). So $\frac{5}{x}$ is a rational expression with restriction $x \neq 0$.