To factorise a cubic polynomial, you can follow a step-by-step approach that involves finding a root and then performing division to obtain a quadratic factor.
Steps to Factorise a Cubic Polynomial
Here's a detailed breakdown of the steps involved in factorising a cubic polynomial, based on the reference provided:
| Step | Description |
|---|---|
| 1 | Find a Root: Identify a root, let's say 'a', of the cubic polynomial. This means that when 'a' is substituted into the polynomial, the result is zero. A potential root can often be found among the prime factors of the constant term of the polynomial. If f(a) = 0, then (x - a) is a factor. |
| 2 | Divide to Find Quadratic Factor: Divide the original cubic polynomial by the linear factor (x - a). The result of this division will be a quadratic polynomial. |
| 3 | Factorise Quadratic Factor: Factorise the quadratic polynomial obtained in the previous step. This can be done using techniques such as factoring by grouping, completing the square, or using the quadratic formula. |
Example
Let's consider a simple example. Suppose you have the cubic polynomial:
f(x) = x3 - 6x2 + 11x - 6
Step 1: Find a Root
By trying factors of -6 (±1, ±2, ±3, ±6), we find that f(1) = 1 - 6 + 11 - 6 = 0. Therefore, x = 1 is a root, and (x - 1) is a factor.
Step 2: Divide to Find Quadratic Factor
Divide x3 - 6x2 + 11x - 6 by (x - 1). This can be done using polynomial long division or synthetic division. The result is x2 - 5x + 6.
Step 3: Factorise Quadratic Factor
Factorise the quadratic x2 - 5x + 6. This factors into (x - 2)(x - 3).
Therefore, the complete factorisation of the cubic polynomial is:
x3 - 6x2 + 11x - 6 = (x - 1)(x - 2)(x - 3)
Additional Considerations
- Rational Root Theorem: The Rational Root Theorem can help narrow down the possible rational roots of the polynomial.
- Synthetic Division: Synthetic division is a faster method for dividing by a linear factor.
- Non-Factorable Quadratics: If the quadratic factor obtained is not factorable using real numbers, the cubic polynomial has only one real root and two complex roots.
