AP Pre-Calculus · Worked Example
AP Pre-Calculus Rational Function Inequality Solved: Where Is h(x) > 0?
A classic AP Pre-Calculus problem (and a technique you'll see again in AP Calculus): given f(x) = x³ − 3x² − 10x and g(x) = x² − 6x + 5, find every interval where h(x) = f(x)/g(x) is positive. Here is the full method — factor, sign chart, answer — plus a 90-second video walkthrough from a SOMATH instructor on the Upper West Side.
The short answer: h(x) > 0 on (-2, 0) ∪ (1, 5) ∪ (5, ∞). To get there, factor both polynomials, identify every value that makes the numerator or denominator zero, build a sign chart with those critical points, and read off the intervals where the product of signs is positive — remembering that x = 1 and x = 5 are excluded from the domain because they make the original denominator zero.
This is one of the most reliably tested techniques in AP Pre-Calculus — and the same skill returns in AP Calculus AB when you analyze where derivatives and second derivatives change sign. Below is the step-by-step solution, a video walkthrough, and a few teaching notes from how we cover this in our SOMATH classes at 226 West 79th Street.
The problem
The function f is given by f(x) = x³ − 3x² − 10x, and the function g is given by g(x) = x² − 6x + 5. Let h be the function given by h(x) = f(x) / g(x). What are all intervals on which h(x) > 0?
- (A) (-2, 0) ∪ (5, ∞)
- (B) (-2, 0) ∪ (1, ∞)
- (C) (-2, 0) ∪ (1, 5) ∪ (5, ∞)
- (D) (-∞, -2) ∪ (0, 1) ∪ (5, ∞)
Video walkthrough (90 seconds)
If you'd rather watch than read, here is the full method in under two minutes.
Watch on YouTube · Subscribe to SOMATH
Step-by-step solution
Step 1: Factor f(x) and g(x)
Always factor first. It exposes every zero, every domain restriction, and every sign flip in one move.
f(x) = x³ − 3x² − 10x
f(x) = x(x² − 3x − 10)
f(x) = x(x − 5)(x + 2)
g(x) = x² − 6x + 5
g(x) = (x − 1)(x − 5)
So h(x) = x(x − 5)(x + 2) / [(x − 1)(x − 5)].
The (x − 5) cancels — but x = 5 must still be excluded from the domain because it makes the original denominator zero. That is a hole in the graph, not a real value of h.
Step 2: Find every critical value
From the numerator: x = -2, 0, 5
From the denominator: x = 1, 5
Critical values in order: -2, 0, 1, 5. These split the number line into five intervals.
Step 3: Build the sign chart
Track the sign of each linear factor across each interval. The sign of h(x) is the product of the signs in each column.
| Factor | (-∞, -2) | (-2, 0) | (0, 1) | (1, 5) | (5, ∞) |
|---|---|---|---|---|---|
| x | − | − | + | + | + |
| x + 2 | − | + | + | + | + |
| x − 5 | − | − | − | − | + |
| x − 1 | − | − | − | + | + |
| h(x) | − | + | − | + | + |
Step 4: Read off the answer
h(x) is positive on the intervals where the bottom row is positive — and we exclude x = 5 because it is not in the domain.
h(x) > 0 on (-2, 0) ∪ (1, 5) ∪ (5, ∞). The correct answer is (C).
School of Math (SOMATH) · UWS: 226 West 79th Street · (646) 668-6151 · schoolofmath.us
Why this technique matters beyond this question
Rational-function sign analysis is one of the highest-yield skills on the AP Pre-Calculus exam — and you will use the same procedure all year long in AP Calculus. Factor, find zeros and undefined points, build a sign chart. The same workflow answers a long list of free-response and multiple-choice questions:
- What is the domain of a rational function? The denominator's zeros are the gaps.
- Where is a polynomial or rational function positive or negative? Sign chart of the function itself.
- Where is f increasing or decreasing? (AP Calc) — sign chart of f'(x).
- Where is f concave up or concave down? (AP Calc) — sign chart of f''(x).
- Inflection points, local extrema, intervals of monotonicity — all sign-chart questions.
Students who can reliably build a clean sign chart finish AP free-response questions faster, lose fewer points to careless arithmetic, and avoid the most common mistake on this kind of problem: forgetting to exclude domain holes.
The three mistakes we see most often
Across our AP Pre-Calculus classes at SOMATH, the same three errors show up on this problem type:
- Cancelling x = 5 and forgetting it. A factor that cancels still creates a hole — the value is excluded from the domain. The correct answer here splits (1, ∞) into (1, 5) and (5, ∞) for exactly this reason.
- Including the zeros in a strict inequality. If the question says h(x) > 0, the zeros of the numerator (x = -2 and x = 0 here) are not in the solution. They make h(x) equal to zero, not greater than zero. Use open intervals.
- Plugging numbers in instead of tracking signs. Computing h(-3) is fine — but tracking the sign of each factor (−, −, −, − ⇒ negative product) is faster and lower-error. Train your eye on signs, not values.
How we teach this at SOMATH
Our AP Pre-Calculus classes meet in small groups at 226 West 79th Street on the Upper West Side. We teach the sign-chart method early, then return to it across every unit — polynomial behavior, rational functions, exponentials, and trig — so the technique is automatic by the time students hit AP Calculus the following year.
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