Understanding the Factor Theorem

You have probably learnt how to factorise a quadratic expression: for example, x2 – 2x – 8 factorises to give (x – 4)(x + 2). This is useful because it enables us to solve the equation x2 – 2x – 8 = 0. How? Because if we use the factorised form we get (x – 4)(x + 2) = 0, and if two numbers multiply to give zero then one of the number must be zero. So either x – 4 = 0, in which case 
x = 4; or x + 2 = 0, in which case x = -2. We can verify these solutions by substituting them into the original equation:

        When x = 4, x2 – 2x – 8 = 42 – 2 × 4 – 8 = 16 – 8 – 8 = 0

        When x = -2, x2 – 2x – 8 = (-2)2 – 2 × (-2) – 8 = 4 + 4 – 8 = 0

These solutions, known as the roots of the equation, also tell us that the graph of y = x2 – 2x – 8 intersect the x-axis at (-2, 0) and (4, 0).

It would be useful if we could factorise cubic expressions and above in the same way, thus enabling us to solve higher degree equations and draw the graphs. (Note: none of these expressions, including quadratics, necessarily factorise – it depends on the numbers – but if they do it saves a lot of head-scratching)! The Factor Theorem is a technique we can use to test whether (ax – b) is a factor of a polynomial. Let me show you how it works.

Consider the cubic P(x) = x3 + 2x2 – 11x – 12. Now, I’ve chosen this because I know it factorises to (x + 1)(x – 3)(x + 4). So:

        P(x) = x3 + 2x2 – 11x – 12 = (x + 1)(x – 3)(x + 4)

Now let’s consider the root x = 3. If I substitute x = 3 into the right hand side I get 4 × 0 × -4 = 0. Therefore I must also get 0 when I substitute x = 3 into the left hand side: 27 + 18 – 33 – 12 = 0. In other words, P(3) = 0. Let’s turn this around: since P(3) = 0, it follows that x – 3 must be a factor. We can generalise this to: if P(a) = 0, then x – a is a factor of P(x). You might like to try this out by proving that x + 2 is a factor of x3 + 3x2 – 4x – 12 by substituting x = -2.

Once you have found a factor, there are various methods for dividing that factor into the polynomial to find the other factors. In this case
                              x3 + 3x2 – 4x – 12 = (x + 2)(x – 2)(x + 3)
but I shall look at these methods in another blog.

So the blue bit above is the factor theorem. Nearly. We can refine it so that we can also test for factors such as (2x – 3). The full factor theorem states that:

                     If P(b/a) = 0, then axb is a factor of P(x)

For example, use the factor theorem to prove that 2x – 3 is a factor of
2x3 + 3x2 – 5x – 6. Do this by substituting x = 1.5 into the polynomial – you should get 0.

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