The Factor Theorem

Introduction

Hopefully an easy ride. Started off with a quadratic equation
x² - 8x - 20 = 0

This works out to

(x + 10)(x -2) = 0

So (x + 10) and (x -2) are factors and we can see either x = -10 and x = 2 are known as solutions. The take away was given we know solution we can reverse engineer factors. So if x = 2 is a solution, (x -2) is a factor

Detail

So here is comes, a long statement

If f(x) is a polynomial and f(a) = 0, then (x-a) is a factor of f(x) - argghhhhhh

The robot translated this into Iain English

If you plug a number into a polynomial and the result is zero, then (x − that number) must be one of its factors.

So an example question might be show that (x+5) is a factor of 2x³ + 15x² + 27x + 10.

To is x + 5 is a factor, it means a solution is x = -5 so

  f(-5) = 2(-5)³ + 15(-5)² + 27(-5)  + 10.
  f(-5) =  -250 + 175 -135 + 10
  f(-5) = 0

Another example

The video suggested when choose a possible solution to look at the factors of the constant so given

 x³ + 4x² - 11x  - 30

So look at the factors of 30, 1,2,3,5,6,10,15,30 and negatives. Using 3 for example

 f(3) = 3³ + 4*3² - 11*3 - 30
 f(3) = 27 + 36 - 33 - 30
      = 0
 (x-3) is a factor

Back to long Division

Once you know one factor, in the is (x-3) we can do long division to find the other.

        x² + 7x + 10
      ----------------------
(x-3) |  x³ + 4x² - 11x - 30
      - (x³ - 3x²)
        --------------------
              7x² - 11x - 30
              7x² - 21x 
              -----------
                    10x - 30
                    10x - 30
                    --------
                          0

So we now have

  x³ + 4x² - 11x  - 30 = (x-3)(x² + 7x + 10)

Where (x-3) is a linear and x² + 7x + 10 is a is a quadratic which we can factorise to

                       = (x-3)(x +5)(x +2)

Another Question

Show that

(2x+1) is a factor of 2x³ + 11x² + 17x  + 6. 

Solving 2x + 1 = 0
2x= -1
x = -1/2

So

   f(-1/2) = 2(-1/2)³ + 11(-1/2)² + 17(-1/2)  + 6. 
           = 2(-1/8) + 11(1/4) + 17(-1/2) + 6
           = -1/4 + 11/4 - 34/4 + 24/4
           = 0