4.3 Rational Expressions, Equations, and Graphs

Rational expressions begin with restrictions

A rational expression is a ratio of polynomials. Because division by zero is undefined, every rational question should begin with the original denominator. The values that make an original denominator zero are non-permissible values. Record them before cancelling factors, before multiplying both sides of an equation, and before using a calculator table.

This order matters because simplification can hide restrictions. If a factor cancels, the simplified expression may look defined at a value where the original expression was not defined. The graph feature at that removed input is a hole, not an ordinary point.

Simplifying rational expressions

The procedure is steady:

1. Factor numerators and denominators completely.
2. State non-permissible values from all original denominators.
3. Cancel common factors, not common terms.
4. Write the simplified expression with the restrictions.
5. Interpret cancelled and uncancelled denominator factors if a graph is involved.

You may cancel (x - 3) from a numerator and denominator, but you may not cancel the x from x + 3. Factors are multiplied pieces; terms are added or subtracted pieces.

Worked example: simplification and graph features

Simplify and interpret:

r(x) = (x^2 - 9)/(x^2 - x - 6)

Factor first:

r(x) = [(x - 3)(x + 3)] / [(x - 3)(x + 2)]

The original denominator is (x - 3)(x + 2), so x != 3 and x != -2. After cancelling (x - 3), the simplified expression is (x + 3)/(x + 2), with the same restrictions from the original expression.

Now interpret the graph. The cancelled factor x - 3 creates a hole at x = 3. The remaining denominator factor x + 2 creates a vertical asymptote at x = -2. The numerator x + 3 creates an x-intercept at x = -3 because that value is allowed.

A complete Math 30-2 answer should not stop at (x + 3)/(x + 2). It should include x != 3 and x != -2, especially in written response.

Multiplying, dividing, adding, and subtracting

Multiplication and division are factor-friendly. Factor everything, change division to multiplication by the reciprocal, record restrictions from the original expression, and cancel common factors.

Addition and subtraction require a common denominator. The diploma trap is failing to distribute the subtraction or failing to multiply the whole numerator. For example:

3/(x - 2) - 1/(x + 1)

has common denominator (x - 2)(x + 1). The combined numerator is 3(x + 1) - 1(x - 2), not 3x + 1 - x - 2. Use brackets until the final simplification.

Solving rational equations

A rational equation contains rational expressions set equal to something. Start with restrictions, then multiply every term by the least common denominator. This clears denominators and usually leaves a linear or quadratic equation. The final check is not optional: any solution that makes an original denominator zero is extraneous.

Example:

3/(x - 2) + 1 = 5/(x - 2)

Restriction: x != 2. Multiply every term by x - 2:

3 + (x - 2) = 5

x + 1 = 5, so x = 4. Since x = 4 does not violate the restriction, it is valid.

Diploma traps

• Cancelling terms instead of factors.
• Writing restrictions after simplifying and missing a cancelled denominator factor.
• Multiplying only some terms by the least common denominator.
• Keeping a solution that makes the original denominator zero.
• Calling every excluded value a vertical asymptote; cancelled factors are holes.

Graphing calculator caution

A calculator graph can make rational behaviour easier to see, but window choice can hide a vertical asymptote or make a hole invisible. The official formula sheet includes graphing calculator window format, which is a reminder that window settings matter. Algebra should confirm what the graph suggests.

Non-permissible values in written response

In written-response work, restrictions should appear before the algebra that removes denominators. This shows the marker that you know which values were never allowed. A clean line such as "Restrictions: x != 2, x != -5" is enough when the restrictions are obvious from factored denominators. If the denominator is not factored, show the factoring step.

When a proposed solution is rejected, state why. For example, if solving produces x = 2 but the original denominator contained x - 2, the final sentence should say that x = 2 is extraneous because it makes an original denominator equal zero. Do not just cross it out. The explanation is part of the mathematics.

Rational graph checklist

For graph questions, organize features in this order: non-permissible values, holes from cancelled factors, vertical asymptotes from remaining denominator factors, intercepts from allowed zeros, and end behaviour or horizontal asymptote. This order mirrors the algebra and prevents a common error: treating a removed factor as a vertical asymptote.

Intercepts and asymptotes together

A rational graph can have several features close together, so separate them before sketching. The x-intercepts come from allowed zeros of the numerator. The y-intercept comes from substituting x = 0, if zero is allowed. Vertical asymptotes come from uncancelled denominator zeros. Holes come from cancelled factors and are excluded points, even when the simplified curve would otherwise pass through them.

A fast sketch should show asymptotes first, then intercepts, then holes. This keeps the graph consistent with the restrictions instead of turning the sketch into a smooth polynomial-looking curve.