A cubic can have one, two or three distinct real roots, and differentiation can often tell you the number without solving the cubic equation. Find the stationary points, calculate their y-values and compare those heights with zero. For a positive-leading-coefficient cubic with two distinct stationary points, three crossings occur when the local maximum lies above the x-axis and the local minimum lies below it.
For example, f(x) = x³ − 3x + k has stationary heights k + 2 and k − 2. It has three distinct real roots when −2 < k < 2, two distinct real roots with a repeated root when k = −2 or 2, and just one real root when k < −2 or k > 2. This guide derives each case, explains the graph, and shows how boundary tangency connects differentiation with polynomial factors.
Root counting is different from solving for exact roots
A real root of f(x) = 0 is an x-coordinate where the graph y = f(x) meets the x-axis. A cubic polynomial has degree three, but its distinct real roots need not number three.
Some cubics cross the x-axis once. Others cross three times. A cubic with a repeated root may have two distinct real x-values even though the roots have total multiplicity three.
When a question asks only “how many real roots?”, a full cubic formula or numerical root approximation may be unnecessary. A graph’s turning behaviour and the signs of its stationary heights can answer the question directly.
This method combines an algebraic function, the derivative and the intermediate value principle: a continuous curve moving from one side of the x-axis to the other must cross it somewhere. Use that reasoning with the derivative’s increasing and decreasing intervals to establish the exact count.
Start with the derivative of the cubic
Let f(x) = x³ − 3x + k, where k is a real constant. Differentiate to obtain f′(x) = 3x² − 3 = 3(x − 1)(x + 1).
Stationary points occur when f′(x) = 0, so the stationary inputs are x = −1 and x = 1. These values do not depend on k because k disappears under differentiation.
The constant k shifts the entire graph up or down without changing where the turning points occur horizontally. That is why examining the turning-point heights is such an efficient route for this family.
A frequent mistake is to solve f(x) = 0 immediately. Differentiation is simpler here, and it reveals the graph structure needed to count intersections before finding any exact root.
Identify the local maximum and local minimum
The derivative 3(x + 1)(x − 1) is positive for x < −1, negative for −1 < x < 1 and positive for x > 1.
Therefore f increases on the far left, decreases between its stationary inputs and increases again on the far right. It has a local maximum at x = −1 and a local minimum at x = 1.
A quick sign chart records +, −, + for the derivative over the three intervals. This gives an increasing–decreasing–increasing graph.
The classifications matter. If a curve’s leading coefficient were negative, its derivative signs would reverse and the left stationary point might instead be a minimum. Here the positive cubic coefficient fixes the orientation and the roles of the two stationary inputs.
Calculate the two stationary heights
At x = −1, f(−1) = (−1)³ − 3(−1) + k = −1 + 3 + k = k + 2. This is the local maximum height.
At x = 1, f(1) = 1 − 3 + k = k − 2. This is the local minimum height.
The root-count question has now become a parameter inequality question: is the maximum above, at or below zero, and is the minimum above, at or below zero?
Because the difference between the two stationary heights is four, the maximum is always strictly higher than the minimum. Their relative position is fixed; changing k moves both heights together by the same vertical amount.
Why three real crossings need opposite stationary signs
For a positive-leading-coefficient cubic, the far-left output tends to negative infinity, while the far-right output tends to positive infinity.
If the local maximum is positive, the graph must cross from negative to positive before or by reaching that maximum. If the local minimum is negative, the graph must then cross from positive to negative between the turning points and finally from negative to positive after the minimum.
The derivative signs ensure exactly one crossing on each monotonic segment when the local maximum is above zero and the local minimum is below zero.
Thus three distinct real roots occur exactly when f(−1) > 0 and f(1) < 0. For the given family, those are k + 2 > 0 and k − 2 < 0. Their intersection is −2 < k < 2.
A complete three-root example at k = 0
Set k = 0. Then f(x) = x³ − 3x, whose local maximum height is 2 and local minimum height is −2.
The stationary heights lie on opposite sides of zero, so there must be three distinct real roots. In this particular case factorisation makes them explicit: x³ − 3x = x(x² − 3).
The roots are x = −√3, x = 0 and x = √3. They lie in three distinct monotonic regions: left of −1, between −1 and 1, and right of 1.
The factorisation confirms the prediction. In less convenient parameter cases, the derivative reasoning can still establish the number of roots even when the exact root values are awkward.
Three roots can occur without neat factorisation
Take k = 1, giving f(x) = x³ − 3x + 1. The maximum height is k + 2 = 3 and the minimum is k − 2 = −1.
Because the maximum is positive and the minimum negative, the equation has exactly three distinct real roots. A simple integer factorisation is not required to make that statement.
Their locations can also be bracketed. Since f(−2) = −1 and f(−1) = 3, one root lies in (−2, −1). Since f(0) = 1 and f(1) = −1, another lies in (0, 1).
Finally, f(1) = −1 and f(2) = 3, so the third lies in (1, 2). The derivative shows that there cannot be an additional root hidden in any of these monotonic intervals.
What happens when the local minimum reaches zero?
At k = 2, the local minimum height k − 2 equals zero. The cubic becomes f(x) = x³ − 3x + 2.
Factorise: f(x) = (x − 1)²(x + 2). The real roots are x = 1, repeated, and x = −2, simple.
There are two distinct real root values, not three. The graph crosses the x-axis at −2 and touches it at the local minimum x = 1 without changing sign there.
The repeated root at x = 1 is consistent with f(1) = 0 and f′(1) = 0. This boundary case separates the three-crossing regime from the one-crossing regime as k increases.
What happens when the local maximum reaches zero?
At k = −2, the local maximum height k + 2 equals zero. The cubic is f(x) = x³ − 3x − 2.
Its factorisation is (x + 1)²(x − 2). Therefore x = −1 is a repeated root and x = 2 is a simple root.
Again there are two distinct real root values. The graph touches the x-axis at its local maximum x = −1, then remains below until crossing at x = 2.
The two boundary values k = −2 and k = 2 are mirror cases. Both give a repeated stationary root, but one tangency occurs at the maximum and the other at the minimum.
Why k greater than two gives one real root
If k > 2, the local minimum height k − 2 is positive. The local maximum is even higher, since its height is k + 2.
After crossing the x-axis somewhere on the increasing left branch, the cubic rises to a positive maximum, descends only to a still-positive minimum and then rises without bound. It cannot cross the axis again.
Therefore there is exactly one real root. It lies to the left of the local maximum at x = −1.
For example, at k = 3, the local minimum is 1 and the maximum is 5. The curve has one negative real root. A graph that showed three crossings for this parameter would contradict the exact stationary-height calculation.
Why k less than negative two also gives one real root
If k < −2, the local maximum height k + 2 is negative. The local minimum is even lower, with height k − 2.
The curve stays below the x-axis through both stationary points. Only on the final increasing branch to the right does it rise from negative values toward positive infinity and cross zero once.
Thus there is exactly one real root, lying to the right of x = 1. For example, k = −3 gives local maximum −1 and local minimum −5.
This mirrors the k > 2 case. Both have one crossing, but the crossing lies on opposite outer branches because the cubic has been shifted in opposite vertical directions.
Summarise the parameter cases precisely
For f(x) = x³ − 3x + k, all real k fall into five mutually exclusive cases.
If k < −2, there is one distinct real root. If k = −2, there are two distinct real roots, one repeated. If −2 < k < 2, there are three distinct real roots.
If k = 2, there are two distinct real roots, one repeated. If k > 2, there is one distinct real root.
A useful way to remember this is through the stationary heights rather than a memorised table. Three roots require maximum above and minimum below zero. Equality produces tangency and a repeated root. When both stationary heights lie strictly on the same side of zero, only one crossing remains.
Use a graph sketch as a consistency check
The function y = x³ − 3x is the baseline shape, with maximum at (−1, 2) and minimum at (1, −2). Adding k moves it vertically without changing its curvature or stationary x-coordinates.
For k = 0, the x-axis cuts the rising left branch, descending middle branch and rising right branch. For k = 2, the minimum rises until it touches the x-axis.
For k > 2, the whole turning region is above the axis, leaving only the left crossing. The opposite happens below k = −2.
A sketch can help readers understand the transitions, but it should reflect the exact values k + 2 and k − 2. Do not infer parameter thresholds only by visually estimating a rough drawing.
The derivative establishes an upper bound on crossings
A cubic derivative is quadratic and therefore has at most two distinct real roots. These divide the real line into at most three intervals on which a non-degenerate cubic is monotonic.
On a strictly monotonic interval, a continuous function can cross a given horizontal level at most once. Thus, with two distinct stationary inputs, the cubic can have at most three distinct real roots.
The fundamental reason our counting method works is the combination of monotonicity and endpoint sign information. The two stationary heights tell us whether each monotonic segment crosses zero.
The polynomial’s degree alone gives an upper bound of three, but the derivative explains when the bound is attained and how the roots are distributed across the graph.
A cubic without two turning points can have one real root
Consider g(x) = x³ + x + 1. Its derivative is g′(x) = 3x² + 1, which is strictly positive for every real x.
The function increases continuously across the entire real line. It tends to negative infinity as x tends to negative infinity and positive infinity as x tends to positive infinity.
Therefore it crosses the x-axis exactly once. There are no local maxima or minima to compare because the derivative never vanishes.
This case shows that the stationary-height test applies when distinct turning points exist. If a cubic has no stationary point, or only a stationary inflection, use monotonicity and end behaviour instead.
A stationary inflection is not a turning point
Take h(x) = x³ + k. Its derivative is h′(x) = 3x², which vanishes at x = 0 but is positive on both sides except at that point.
The function remains increasing through x = 0. There is a horizontal tangent but no change from increasing to decreasing or vice versa.
The equation x³ + k = 0 has exactly one real root for every real k, namely x = the real cube root of −k. If k = 0, that root is a triple root at the stationary inflection.
A student who counts every zero derivative as a maximum or minimum may incorrectly predict three real roots. Inspect sign changes in the derivative, not just where it equals zero.
Extend the method to a wider cubic family
Consider f(x) = x³ − 12x + k. Its derivative is f′(x) = 3x² − 12 = 3(x − 2)(x + 2), so the stationary inputs are −2 and 2.
At x = −2, the local maximum is f(−2) = −8 + 24 + k = 16 + k. At x = 2, the local minimum is f(2) = 8 − 24 + k = k − 16.
Three distinct roots therefore occur when 16 + k > 0 and k − 16 < 0, giving −16 < k < 16.
At k = ±16 there is a repeated root and two distinct real values; outside that range there is one real root. The thresholds change, but the reasoning is identical to the x³ − 3x + k example.
Check the boundary factorisation for the wider family
At k = 16, the wider cubic becomes x³ − 12x + 16. It factorises as (x − 2)²(x + 4).
Thus x = 2 is a repeated root at the local minimum and x = −4 is the other real root. At k = −16, the cubic becomes x³ − 12x − 16 = (x + 2)²(x − 4).
There the repeated root is x = −2 at the local maximum, with the other root x = 4. Both results match the stationary-height threshold analysis.
Factorisation is a useful independent check at the boundary, but it is not needed to find the full parameter interval −16 < k < 16 for three distinct roots.
What if the cubic has a negative leading coefficient?
A cubic with negative leading coefficient tends to positive infinity on the far left and negative infinity on the far right. Its orientation is the reverse of a positive-leading cubic.
For example, −x³ + 3x + k has derivative −3x² + 3. At x = −1 it has a local minimum of k − 2, while at x = 1 it has a local maximum of k + 2.
Three distinct real roots still occur when the local maximum is above zero and the local minimum is below zero. That gives −2 < k < 2.
The general principle does not change, but one must identify which stationary point is the maximum and which is the minimum rather than assuming the left one is always the maximum.
Why a repeated root creates tangency
Suppose f(r) = 0 and f′(r) = 0 for a polynomial. If r is a root of multiplicity at least two, both conditions hold. Conversely, for a polynomial, simultaneous vanishing of f and its derivative indicates a repeated root.
For x³ − 3x + 2, the value x = 1 satisfies both equations and the factorisation contains (x − 1)². The graph meets the x-axis with a horizontal tangent there.
At x = −2, the same polynomial has a simple root. Its derivative is positive rather than zero, so the graph crosses the axis with nonzero gradient.
This comparison explains why the boundary cases have two distinct real roots instead of three: two algebraic roots have merged at a stationary contact point.
The cubic discriminant offers an advanced cross-check
For the depressed cubic x³ + px + q, the cubic discriminant is −4p³ − 27q². Its sign distinguishes three distinct real roots from a repeated-root case or one real root.
For x³ − 3x + k, we have p = −3 and q = k. The discriminant becomes 108 − 27k² = 27(4 − k²).
It is positive exactly when −2 < k < 2, zero when k = ±2 and negative outside. This agrees with our derivative and stationary-height analysis.
The cubic-discriminant formula is optional higher-level enrichment. Students need not memorise it to solve these questions; the derivative gives an accessible geometric explanation of the same thresholds.
A single numerical sample never proves the root count
Checking that f(0) is positive or negative is not sufficient to count all real roots of a cubic. A cubic can cross the x-axis more than once even when one sampled value is above zero.
For example, f(x) = x³ − 3x + 1 has f(0) = 1, but it has three real roots because its local maximum is positive and its local minimum is negative.
To prove the count, use the monotonic intervals determined by f′ and the signs at the stationary points, together with the cubic’s end behaviour.
Numerical substitutions can bracket roots and check a diagram. They cannot replace the complete reasoning about how many times the continuous function is able and required to cross the axis.
Common errors and their targeted repairs
One student finds stationary x-values but forgets to evaluate the corresponding heights. Another checks those heights but does not classify which is the maximum and which is the minimum.
A third counts a repeated root twice when asked for distinct roots. A fourth assumes that a zero derivative must be a turning point, failing on a stationary inflection.
These errors require different repairs. Use a simple sign chart for derivative behaviour, boundary factorisations for repeated roots, and the example x³ + k for a stationary inflection.
A changed cubic family should be used to check whether the learner can repeat the reasoning independently without being told the critical parameter thresholds.
A six-stage examination solution
First differentiate the cubic. Second solve f′(x) = 0 and decide whether there are two distinct stationary points, one stationary input or none.
Third classify the stationary points using derivative signs. Fourth calculate their exact y-values as expressions in any parameter. Fifth compare the maximum and minimum with zero and solve the resulting inequalities.
Sixth address equality boundaries separately, using factorisation or f(r) = f′(r) = 0 to identify repeated roots. State the number of distinct real roots clearly.
This solution is usually much shorter than trying to solve a parameterised cubic equation explicitly and then inferring how many real answers it happens to have.
Practice questions with checked answers
Question one: for x³ − 3x + k = 0, find k for exactly three distinct real roots. Answer: −2 < k < 2.
Question two: find k for exactly two distinct real roots. Answer: k = −2 or 2, with a repeated root.
Question three: find k for exactly one distinct real root. Answer: k < −2 or k > 2.
Question four: for x³ − 12x + k = 0, find k for three distinct real roots. Answer: −16 < k < 16.
Question five: for g(x) = x³ + x + 1, determine the number of distinct real roots. Answer: one, because g′(x) = 3x² + 1 > 0.
Question six: factor x³ − 3x + 2. Answer: (x − 1)²(x + 2), with roots 1 and −2 as distinct values.
Explain each root count using stationary heights, monotonicity or a boundary factorisation rather than a calculator graph alone.
Syllabus relevance and companion guides
This method connects differentiation, stationary points, polynomial roots, parameter inequalities and graph interpretation. The exact level of cubic root-count problems and the optional discriminant extension should be selected according to the learner’s taught examination syllabus.
For Singapore, check the official SEAB lists for 2026 O-Level Additional Mathematics 4049, 2027 SEC G3 K341 or 2027 SEC G2 K232, as applicable.
For related learning, read How to Reconstruct a Cubic from Two Stationary Points and One Point, How to Reconstruct a Cubic from a Stationary Point and a Root and Why Tangency Creates a Repeated Root.
The eduKateSG Additional Mathematics archive offers further resources. Parents may enquire about suitable maximum three-student tutorials near Sixth Avenue MRT. The educational objective is to predict the number of real solutions from a curve’s exact behaviour and explain the thresholds where that number changes. Properly taught kids shine a bright light into the future.