Rational Function Zeros

Factor both the numerator P(x) and denominator Q(x) completely. For example, if R(x) = (x² − x − 6)/(x² − 4), factor to get R(x) = (x − 3)(x + 2) / [(x − 2)(x + 2)]. Full factoring reveals any common factors between numerator and denominator, which correspond to holes rather than zeros.

If any linear factor appears in both the numerator and denominator, that factor corresponds to a hole (removable discontinuity), not a zero. In the example, (x + 2) is common, so x = −2 is a hole. After cancellation: R(x) = (x − 3)/(x − 2), with a hole at x = −2. Do not include x = −2 as a zero.

Solve P_simplified(x) = 0. In the example, set x − 3 = 0, giving x = 3. Confirm that x = 3 does not make the simplified denominator zero: x − 2 evaluated at x = 3 gives 1 ≠ 0. Therefore x = 3 is a genuine zero.

The multiplicity of a zero in the simplified numerator determines the graph's behaviour at that x-intercept. If the factor (x − a) appears to an odd power, the graph crosses the x-axis at x = a. If it appears to an even power, the graph touches the x-axis and turns back. For example, R(x) = (x − 1)²(x + 3) / (x − 4) has a zero at x = 1 of multiplicity 2 (graph touches, does not cross) and a zero at x = −3 of multiplicity 1 (graph crosses).

Substitute the zero back into the original (unsimplified) function to confirm R(zero) = 0 and the function is defined there. You can also construct a sign chart of R(x) around each zero: if R(x) changes sign across x = a, the graph crosses there; if R(x) keeps the same sign, the graph touches and turns.

• R(x) — Rational function (dimensionless): The function whose zeros are being found; it is the ratio of two polynomials.
• P(x) — Numerator polynomial (dimensionless): The polynomial in the numerator; its roots are candidates for zeros of R(x).
• Q(x) — Denominator polynomial (dimensionless): The polynomial in the denominator; it must not equal zero at any zero of R(x), otherwise a hole or asymptote exists instead.

Problem. Find all zeros of the rational function R(x) = (2x³ − 8x) / (x² − 5x + 6). State the x-intercepts and identify any holes.

Answer. The zeros (x-intercepts) of R(x) are x = 0 and x = −2. The graph crosses the x-axis at both points. There is a hole at x = 2 (not a zero). There is a vertical asymptote at x = 3.

• Rational Function Asymptotes — Asymptotes and zeros together define the key structural features of a rational function's graph. While zeros tell you where the graph meets the x-axis, vertical asymptotes tell you where the denominator is zero (and no zero of R(x) can occur), and horizontal or oblique asymptotes describe end behaviour.
• Holes in Rational Graphs — Holes arise from common factors in the numerator and denominator — exactly the factors you must cancel before identifying true zeros. Confusing holes with zeros is the most common error in this topic, so understanding both concepts together is essential.
• Polynomial End Behaviour — The zeros of the numerator polynomial are found using polynomial techniques. Understanding polynomial structure — including how factoring reveals roots — is directly needed to locate the zeros of rational functions.
• Equivalent Algebraic Forms — Finding zeros requires rewriting rational functions in factored form. The concept of equivalent algebraic forms explains why cancellation of common factors is valid and what it means for a simplified expression to represent the same function except at removed points.
• Polynomial Symmetry — Symmetry of the numerator polynomial can predict the placement of zeros: an even numerator polynomial has zeros symmetric about x = 0, which can simplify the search for zeros of the rational function. Recognising symmetry is a useful shortcut when analysing rational functions.
• Polynomial & Rational Modeling — In real-world models using rational functions (e.g., drug concentration over time, cost-efficiency functions), the zeros of the model have direct physical meaning — such as the time when a concentration returns to zero. Interpreting zeros in context is a key modeling skill.