Cubic Polynomials
Understand the shape of cubic curves, find roots and turning points, and learn to sketch cubic functions.
The Shape of Cubic Functions
A cubic function has the general form y = ax³ + bx² + cx + d where a ≠ 0. It is a polynomial of degree 3.
The basic cubic y = x³ has an S-shaped curve that passes through the origin. The sign of a determines the end behaviour:
End Behaviour of Cubics
Positive leading coefficient
Falls left, rises right
Negative leading coefficient
Rises left, falls right
A cubic function can have 1 or 3 real roots (x-intercepts) and may have 0 or 2 turning points (local maximum and minimum).
Finding Roots of Cubics
To find the roots (x-intercepts), set y = 0 and solve. Common strategies include:
- Common factor: Factor out x if d = 0, e.g., x³ - 4x = x(x² - 4) = x(x-2)(x+2).
- Factor theorem: If P(a) = 0, then (x - a) is a factor. Test simple values like ±1, ±2, etc.
- Long division or synthetic division: After finding one factor, divide to obtain a quadratic.
Factor Theorem
For a polynomial P(x), if P(a) = 0, then (x - a) is a factor of P(x). This works for cubics and all polynomials.
Multiplicity of Roots
A root with multiplicity 1 (single root) — the curve crosses the x-axis.
A root with multiplicity 2 (double root) — the curve touches and turns at the x-axis.
A root with multiplicity 3 (triple root) — the curve has a point of inflection on the x-axis.
Sketching Cubic Functions
To sketch a cubic, identify these key features:
- Sign of a: Determines the end behaviour (falls-rises or rises-falls).
- Y-intercept: Set x = 0 to find the point (0, d).
- X-intercepts: Factor and solve P(x) = 0.
- Turning points: Find where P'(x) = 0 (if using calculus) or use the shape of the factored form.
- Behaviour at roots: Does the curve cross, touch, or have an inflection at each root?
Example Shapes
3 distinct roots
y = (x+2)(x)(x-3)
Crosses x-axis 3 times
1 single + 1 double root
y = (x-1)²(x+2)
Crosses once, touches once
1 real root only
y = x³ + x + 1
Crosses x-axis once
Key Vocabulary
Cubic Function
A degree-3 polynomial of the form y = ax³ + bx² + cx + d, where a ≠ 0.
Turning Point
A point where the function changes from increasing to decreasing (local max) or vice versa (local min).
Point of Inflection
A point where the curve changes concavity — from concave up to concave down or vice versa.
Multiplicity
The number of times a root appears as a factor. It determines how the curve behaves at that root.
Worked Examples
Factorise P(x) = x³ - 7x + 6.
Step 1: Test x = 1: P(1) = 1 - 7 + 6 = 0. So (x - 1) is a factor.
Step 2: Divide: x³ - 7x + 6 ÷ (x - 1) = x² + x - 6.
Step 3: Factor the quadratic: x² + x - 6 = (x + 3)(x - 2).
Answer: P(x) = (x - 1)(x + 3)(x - 2). Roots at x = -3, 1, and 2.
Sketch y = -x³ + 4x.
Step 1: Factor: y = -x(x² - 4) = -x(x - 2)(x + 2).
Step 2: Roots at x = -2, 0, and 2 (all single roots, so the curve crosses at each).
Step 3: Leading coefficient a = -1 < 0, so the curve rises left and falls right.
Answer: The cubic crosses the x-axis at (-2, 0), (0, 0), and (2, 0), rising on the left and falling on the right, with two turning points between the roots.
Describe the behaviour of y = (x - 1)²(x + 3) at each root.
Step 1: Root x = 1 has multiplicity 2 (double root) — the curve touches and turns at x = 1.
Step 2: Root x = -3 has multiplicity 1 (single root) — the curve crosses the x-axis at x = -3.
Step 3: Leading coefficient is positive (expanding gives x³ + ...), so the curve falls left and rises right.
Answer: The curve crosses the x-axis at x = -3, turns and touches the x-axis at x = 1, with y-intercept at (0, (0-1)²(0+3)) = (0, 3).
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
How many turning points can a cubic function have at most?
Question 2
If P(x) = x³ + 2x² - 5x - 6, what is P(2)?
Question 3
The graph of y = (x - 1)³ at x = 1 will:
Question 4
What is the y-intercept of y = 2x³ - x² + 3x - 4?
Question 5
If the leading coefficient of a cubic is negative, the end behaviour is:
Key Concepts Summary
- ●A cubic function has the form y = ax³ + bx² + cx + d and produces an S-shaped curve.
- ●Cubics have 1 or 3 real roots and at most 2 turning points.
- ●The Factor Theorem states that if P(a) = 0, then (x - a) is a factor.
- ●The multiplicity of a root determines whether the curve crosses, touches, or inflects at the x-axis.
- ●End behaviour is determined by the sign of a: positive means falls-left/rises-right; negative means rises-left/falls-right.