Using the Factor Theorem to Solve Cubic Polynomial Problems

Understanding how to apply the factor theorem is a fundamental skill for A Level Maths students tackling polynomial equations. Let’s explore a typical exam-style question and see how we can use the theorem to prove a given result.

The Problem

Given the function:

f(x) = 4x^{3} + 5x^{2} - 10x + 4ap

where a is a positive constant, and knowing that (x-a) is a factor of f(x), we are asked to show that:

a\left(4a^{2} + 5a - 6 \right) = 0

Step-by-Step Solution

1. Using the Factor Theorem

The factor theorem states that if (x-a) is a factor of f(x) then f(a) = 0

Let’s substitute x = a into the function:

f(a) = 4a^{3} + 5a^{2} - 10a + 4a

Notice that the last two terms can be combined:

-10a + 4a = -6a

So,

f(a) = 4a^3 + 5a^2 - 6a

2. Setting f(a) = 0

Since (x-a) is a factor,

4a^3 + 5a^2 - 6a = 0

3. Factorising

We can factor a from each term:

a(4a^2 + 5a - 6) = 0

Thus, we have shown the required result.

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Discussion

This classic use of the factor theorem not only checks your understanding of polynomial roots and algebraic manipulation, but also demonstrates the importance of careful substitution and factorisation. In exam situations, always look out for opportunities to factorise and simplify expressions after substitution.

Practising these types of problems is crucial for mastering polynomial algebra at A Level. If you want more guided practice and detailed explanations, an Online A Level Maths Revision Course can be extremely beneficial, offering structured revision and expert support.