Question

Using the factor theorem factorise it completely
X cube -7 x square+14x -8

Answer 1

see solution in attachements ......

Answer 2

The factor theorem states that if x−a is a factor of some polynomial f(x), then f(a)=0. The proof of this is very straightforward. Try it as an exercise.

To find factors of your cubic,

f(x)=x3−x2−14x+24

We will try to find values of x for which f(x)=0. Supposing that the polynomial can be neatly factored into integer factors x−a, x−b, and x−c, we know that abc=24, so it is beneficial to try positive and negative factors of 24.

Let's try 1, the smallest:

f(1)=13−12−14⋅1+24=10

Hmm, no luck. Next let's try −1:

f(1)=(−1)3−(−1)2−14(−1)+24=36

Again we have not reached zero. Next let's try 2:

f(2)=23−22−14⋅2+24=0

We've found a solution! This means that x−2 is a factor of the cubic. To find the remaining factors, we can do polynomial division to find some quadratic x2+bx+c so that f(x)=(x−2)(x2+bx+c).