How To Factorise A Cubic Function

How to Factorise a Cubic Function: A Comprehensive Guide

Factorising cubic functions can seem daunting, but with a systematic approach and understanding of several techniques, it becomes manageable. This comprehensive guide will walk you through various methods, from simple inspection to employing more advanced strategies like the Rational Root Theorem and polynomial long division. We'll also explore how to handle different scenarios and interpret the results. By the end, you'll be confident in your ability to factorise a wide range of cubic functions.

Understanding Cubic Functions

Before diving into the methods, let's establish a foundational understanding. A cubic function is a polynomial function of degree three, meaning the highest power of the variable (typically x) is 3. The general form is:

f(x) = ax³ + bx² + cx + d

where a, b, c, and d are constants, and a is not equal to zero. Factorising this function means expressing it as a product of its linear factors (factors of the form (x - r), where r is a root).

Method 1: Simple Inspection (Common Factor Extraction)

The simplest method is to look for common factors among the terms of the cubic function. This is best suited for cubic expressions where a common factor is readily apparent.

Example:

Factorise: f(x) = x³ + 2x² + x

Notice that x is a common factor in all terms. We can extract it:

f(x) = x(x² + 2x + 1)

The quadratic expression in the parentheses can be further factorised:

f(x) = x(x + 1)(x + 1) = x(x + 1)²

Therefore, the completely factorised form of the cubic function is x(x + 1)².

Method 2: The Rational Root Theorem

The Rational Root Theorem provides a systematic way to identify potential rational roots (roots that are rational numbers – fractions or integers) of a polynomial equation. This is a crucial step before employing other techniques like polynomial long division.

The Theorem: If a polynomial equation with integer coefficients has a rational root p/q (where p and q are integers and q ≠ 0), then p is a factor of the constant term (d) and q is a factor of the leading coefficient (a).

Example:

Factorise: f(x) = 2x³ - 5x² - 4x + 3

1. Identify potential rational roots: The factors of the constant term (3) are ±1 and ±3. The factors of the leading coefficient (2) are ±1 and ±2. Therefore, the potential rational roots are ±1, ±3, ±1/2, ±3/2.

2. Test the potential roots: We substitute each potential root into the function. If f(x) = 0, then we've found a root.

   • f(1) = 2(1)³ - 5(1)² - 4(1) + 3 = 0 => (x - 1) is a factor
   • f(-1) = 2(-1)³ - 5(-1)² - 4(-1) + 3 = 0 => (x+1) is not a factor
   • f(3) = 2(3)³ - 5(3)² - 4(3) + 3 = 12 ≠ 0
   • f(-3) = 2(-3)³ - 5(-3)² - 4(-3) + 3 = -72 ≠ 0
   • f(1/2) = 2(1/2)³ - 5(1/2)² - 4(1/2) + 3 = 0 => (2x-1) is a factor
   • f(-1/2) =2(-1/2)³ - 5(-1/2)² - 4(-1/2) + 3 ≠0
   • f(3/2) = 2(3/2)³ - 5(3/2)² - 4(3/2) + 3 = 0 => (2x-3) is a factor

3. Polynomial Long Division: Once we've found a root (e.g., x = 1), we use polynomial long division to divide the cubic function by (x - 1). This will yield a quadratic expression.

   2x³ - 5x² - 4x + 3 ÷ (x - 1) = 2x² - 3x - 3

4. Factor the Quadratic: Now, we need to factor the resulting quadratic (2x² - 3x - 3). This may not factorise neatly using integers, so we might need to use the quadratic formula:

   x = [-b ± √(b² - 4ac)] / 2a = [3 ± √(9 - 4(2)(-3))] / 4 = [3 ± √33] / 4

Therefore, the complete factorisation is:

f(x) = (x - 1)(2x - 1)(x-3/2)

or: (x-1)(2x-1)(2x-3)/2

Method 3: Polynomial Long Division (with a known factor)

If you already know one factor of the cubic function (perhaps through observation or using other methods), you can use polynomial long division to find the remaining quadratic factor.

Example:

Let's assume we know (x + 2) is a factor of f(x) = x³ + 3x² - 4x - 12.

Perform polynomial long division:

(x³ + 3x² - 4x - 12) ÷ (x + 2) = x² + x - 6

The quotient is a quadratic expression (x² + x - 6), which can be further factorised:

x² + x - 6 = (x + 3)(x - 2)

Therefore, the complete factorisation is: f(x) = (x + 2)(x + 3)(x - 2).

Method 4: Sum and Difference of Cubes

This method applies specifically to cubic expressions that are in the form of a sum or difference of cubes:

• Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
• Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)

Example:

Factorise: f(x) = 8x³ - 27

This is a difference of cubes, where a = 2x and b = 3:

f(x) = (2x - 3)(4x² + 6x + 9)

The quadratic factor (4x² + 6x + 9) cannot be factorised further using real numbers. It has complex roots.

Method 5: Grouping (for certain types of cubic expressions)

Sometimes, a cubic expression can be factorised by grouping terms. This is similar to factoring by grouping in quadratic expressions. It works best when you can identify pairs of terms that share a common factor.

Example:

Factorise: f(x) = x³ + 2x² - 9x - 18

Group the terms:

f(x) = (x³ + 2x²) + (-9x - 18)

Factor out common factors from each group:

f(x) = x²(x + 2) - 9(x + 2)

Notice that (x + 2) is a common factor:

f(x) = (x + 2)(x² - 9)

The quadratic factor (x² - 9) is a difference of squares:

f(x) = (x + 2)(x + 3)(x - 3)

Handling Cubic Functions with Complex Roots

Not all cubic functions have three real roots. Some may have complex roots (involving the imaginary unit i, where i² = -1). These complex roots always appear in conjugate pairs (a + bi and a - bi).

When you encounter a quadratic factor that doesn't factorise further using real numbers (as in the difference of cubes example), you can use the quadratic formula to find the complex roots.

Interpreting the Factorisation

Once you've successfully factorised a cubic function, the factors reveal valuable information:

• Roots: The values of x that make f(x) = 0 are the roots (or zeros) of the function. They are the values obtained by setting each factor equal to zero and solving for x.
• x-intercepts: The real roots correspond to the x-intercepts of the graph of the cubic function.
• Multiplicity of Roots: If a factor appears more than once, the corresponding root has a higher multiplicity. For example, in f(x) = x(x + 1)², the root x = -1 has a multiplicity of 2. This means the graph touches the x-axis at x = -1 instead of crossing it.

Advanced Techniques (Beyond the Scope of this Introduction)

For more complex cubic functions, more advanced techniques might be necessary, such as:

• Numerical methods: For functions that are difficult to factorise algebraically, numerical methods (like Newton-Raphson) can approximate the roots.
• Cardano's method: This historical method provides a formula for solving cubic equations, but it is often cumbersome and impractical for most cases.

This guide provides a comprehensive introduction to factorising cubic functions. By mastering these methods and understanding the underlying concepts, you will be equipped to tackle a variety of cubic expressions and gain valuable insights into their properties. Remember to practice regularly and work through various examples to solidify your understanding.