The polynomial p(x) is given by p(x) = x^3 – 5x^2 – 8x + 48 (a) (i) Use the Factor Theorem to show that x + 3 is a factor of p(x). [2 marks] (ii) Express p(x) as a product of three linear factors. [3 marks]

(a) (i) Use the Factor Theorem to show that x + 3 is a factor of p(x).

The Factor Theorem is derived from the remainder theorem. We know from the remainder theorem that by doing p(x)/(x – a) then we get a remainder r = p(a) which is p(x) evaluated at x=a. So for x – a to be a factor of p(x) then p(a) = 0. Using this logic, we can now answer the question.

For x + 3 to be a factor of p(x) we must evaluate p(-3). So we get p(-3) = (-3)3^3 – 5(-3)^2 – 8(-3) + 48 = -27 – 45 + 24 + 48 = 0Therefore, x +3 is a factor of p(x). The 2 marks come from evaluating p(-3) and coming to the conclusion that p(-3) = 0 so x + 3 is a factor.

(ii) Express p(x) as a product of three linear factors.

From the previous question we already know that (x+3) is a factor of p(x) = x^3 – 5x^2 – 8x + 48. Firstly, we want to figure out what we need to multiply x + 3 with to get p(x). This can be done in two ways: inspection or long division.

Let’s start with inspection. We have p(x) = x^3 – 5x^2 – 8x + 48 = (x+3)(ax^2 + bx + c) = ax^3 + bx^2 + cx + 3ax^2 + 3bx + 3c = ax^3 + (3a + b)x^2 + (3b + c)x + 3c.

By looking at the coefficient for x^3 we know that a must be equal to 1. We can also see that the last term c must be equal to 16 as 3c = 48. Now all we are left with is the coefficient b. Consider the coefficient of x after we multiply the brackets out. We have 3b + c = -8. Be very careful here ensuring that the coefficients have the correct sign before. As we know that c is 16 we get 3b = -24 so b = -8. This gives us p(x) = x^3 – 5x^2 – 8x + 48 = (x+3)(x^2 – 8x + 16). It is not over yet.

Always look at the last bracket to see if you can factorise this bracket too which we can. So we get (x^2 – 8x + 16) = (x – 4)(x – 4). Using this we now have a p(x) in terms of 3 linear factors.

p(x) = (x + 3)(x – 4)(x – 4)

The marks are given here are given by using the inspection method. Getting the correct quadratic for ax^2 + bx + c and finally factorising that quadratic and showing p(x) in terms of the 2 linear factors.

Always make sure to double check your working. In this instance you can just multiply out all three brackets and see how you get the original p(x) = x^3 – 5x^2 – 8x + 48.

Unfortunately, it is very tricky to show the long division method here so I would provide a handwritten document showing the methodology instead.