The factor theorem

When factorising polynomials, for example factorise

    \[{x^3} + 3{x^2} - 33x - 35{\text{ \hspace{2em} or \hspace{2em}  }}2{x^4} + 3{x^3} - 19{x^2} - 27x + 18\]

or anything of the form

    \[{a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + ............ + {a_2}{x^2} + {a_1}x + {a_0}\]

It helps to know the factor theorem.

You will be expected to factorise cubic polynomials such as

    \[{x^3} + 3{x^2} - 33x - 35\]

So what is the factor theorem?

  • {{\text{If }}f(x){\text{ is a polynomial and }}f(p) = 0{\text{ , then }}x - p{\text{ is a factor of }}f(x)\text{. }

In the video tutorial I show you why this is so.

Showing that x-1 is a factor of a cubic polynomial

Now that we have the factor theorem, expect to be asked something along the lines of the following examples which you may want to try before looking at the video worked solutions.

Example in the video

Show that x-1 is a factor of:

  1. 2{x^3} - 3{x^2} - x + 2

 

Factorising a cubic polynomial

Examples in the video

Factorise:

  1. 2{x^3} - 3{x^2} - 11x + 6

Method 1

Method 2

Finding constants in a polynomial given the factors

In this tutorial you are shown how to find constants in a given polynomial when you are given some of the factors.

Example in the video

  1. If {x^3} + 2{x^2} + ax + b has factors x+1 and x-2. Find a and b.




2018-08-06T15:40:22+00:00
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