Factorisation of Cubic Polynomials: Explained with Examples

A cubic polynomial has the form ax cubed + bx squared + cx + d, where a is not zero. It is called cubic because the highest power of the variable is 3. Factorising a cubic means writing it as a product of a linear factor and a quadratic factor, and then factorising the quadratic further if possible.

This matters because a cubic can have up to three roots, and factorisation reveals all of them at once. Cubics describe volumes, many engineering systems, and economic models, so being able to break them apart is a genuinely useful skill beyond exams.

Key Concepts

• Degree: the highest power of the variable. A cubic has degree 3.
• Root (or zero): a value of x that makes the polynomial equal zero. Each linear factor (x - r) corresponds to a root r.
• Rational root theorem: any rational root p/q of a cubic with integer coefficients must have p dividing the constant term d and q dividing the leading coefficient a.
• Leading coefficient: the coefficient a of the x cubed term.

How It Works

1. List possible rational roots: factors of the constant term divided by factors of the leading coefficient. For integer-coefficient cubics, just try the factors of the constant term: plus or minus 1, 2, 3, and so on.
2. Substitute each candidate into the polynomial until you find one that gives zero. Call it r.
3. Then (x - r) is a factor. Divide the cubic by (x - r) using grouping or polynomial division to get a quadratic.
4. Factorise the resulting quadratic by the usual methods.
5. Multiply all the factors together mentally to verify the original cubic.

Worked Example

Factorise x cubed - 6x squared + 11x - 6.

1. Possible integer roots are factors of -6: plus or minus 1, 2, 3, 6.
2. Test x = 1: 1 - 6 + 11 - 6 = 0. So x = 1 is a root and (x - 1) is a factor.
3. Divide out (x - 1): x cubed - 6x squared + 11x - 6 = (x - 1)(x squared - 5x + 6).
4. Factorise the quadratic: x squared - 5x + 6 = (x - 2)(x - 3).
5. Final answer: (x - 1)(x - 2)(x - 3).

Verify the other roots: at x = 2, 8 - 24 + 22 - 6 = 0; at x = 3, 27 - 54 + 33 - 6 = 0. All three check out.

Common Misconceptions

• Every cubic factorises neatly with integers. Wrong. Some cubics have irrational or complex roots and cannot be factorised with integers at all, though every cubic with real coefficients has at least one real root.
• You only need to test positive numbers. Wrong. Roots can be negative, so test both signs of each factor of the constant term.
• Finding one root finishes the job. Wrong. After dividing out (x - r) you still have a quadratic to factorise, which may yield two more roots.

Real-World Applications

• Volume problems, such as designing a box with a given volume, lead to cubic equations whose roots give the possible dimensions.
• In engineering, characteristic equations of systems are often cubic, and their roots determine whether the system is stable.
• In economics, cubic cost and revenue models are factorised to find break-even production levels.

To recap: test factors of the constant term to find one root, divide out its linear factor, then factorise the remaining quadratic. Verify every root by substitution, and remember that not every cubic has neat integer factors.