Factorising cubic equations can be a daunting task, but with the right approach, it can be broken down into manageable steps. By understanding the underlying principles and applying systematic methods, even complex cubic equations can be factorised with ease. This guide will provide a comprehensive overview of the various techniques used to factorise cubic equations, empowering you to tackle these algebraic challenges with confidence.
One of the most commonly used methods for factorising cubic equations is the Rational Root Theorem. This theorem states that if a rational number p/q is a root of a polynomial equation with integer coefficients, then p must be a factor of the constant term and q must be a factor of the leading coefficient. By systematically testing potential rational roots based on this theorem, it is possible to identify roots and subsequently factorise the cubic equation.
When the Rational Root Theorem is not applicable or does not yield the desired result, other methods such as synthetic division, grouping, and completing the cube can be employed. Synthetic division involves dividing the cubic polynomial by a linear factor (x – a) to determine if (x – a) is a factor of the polynomial. Grouping involves rewriting the cubic polynomial as a sum or difference of two quadratic expressions, which can then be factorised using the quadratic formula. Completing the cube involves transforming the cubic polynomial into the form (x + a)^3 + b, which can be easily factorised into its linear and quadratic factors
Using a Graph to Guide Factorisation
When you have a cubic equation, y = f(x), you can use a graph of the equation to help you factorise it.
Examining the Graph
First, plot the graph of the equation. Look for the following features:
- Identifiable shapes (e.g. parabolas, lines)
- Points where the graph crosses the x-axis (x-intercepts)
- Maximum and minimum points (turning points)
Identifying the x-intercepts
x-intercepts are points where the graph crosses the x-axis. Each x-intercept represents a root of the equation, where f(x) = 0. If the roots are rational numbers, you can find them by inspection or using the Rational Root Theorem.
Example
Consider the equation y = x3 – 3x2 – 4x + 12. The graph of the equation has x-intercepts at x = 2, x = 3, and x = -2. Therefore, the equation can be factorised as: y = (x – 2)(x – 3)(x + 2).
Dealing with Irrational Roots
If the roots are irrational numbers, you can use the graph to estimate their values. Zoom in on the x-intercepts to find the approximate coordinates of the roots.
Factorisation
Once you have identified the roots, you can factorise the equation. Each root represents a linear factor of the equation. Multiply these factors together to obtain the complete factorisation.
Table of Factors and Roots
Root Factor x = 2 (x – 2) x = 3 (x – 3) x = -2 (x + 2) Therefore, y = (x – 2)(x – 3)(x + 2).
How to Factorise Cubic Equations
Factoring cubic equations can be a challenging task, but it is a necessary skill for anyone who wants to solve these types of equations. Here is a step-by-step guide on how to factorise cubic equations:
- Begin by finding the roots of the equation. To do this, you can use the Rational Root Theorem or synthetic division.
- Once you have found the roots, you can use them to factorize the equation. To do this, simply multiply the roots together to get the coefficient of x^2, and then add the roots together to get the constant term.
- Finally, you can use the coefficients to write the factorised form of the equation.
People Also Ask
How to find the roots of a cubic equation?
There are a few different methods that you can use to find the roots of a cubic equation. One common method is the Rational Root Theorem, which states that the only possible rational roots of a polynomial equation are factors of the constant term divided by the leading coefficient.
Another method that you can use is synthetic division. This method is a simple and efficient way to find the roots of a polynomial equation.
How to factorise a cubic equation by grouping?
To factorise a cubic equation by grouping, you first need to group the terms of the equation into two groups: (x^2 + bx + c) and (ax + d). Once you have grouped the terms, you can factor out the greatest common factor from each group. Then, you can use the distributive property to rewrite the equation as a product of two binomials.
How to solve a cubic equation using the quadratic formula?
You cannot use the quadratic formula to solve a cubic equation. The quadratic formula only works for equations of degree 2.
