Factor Theorem:

 

F(x) has x=a as a root when f(a) = 0 then (x-a) is a factor of f(x).

Follow examples shown below.

 

 

Example 1:

(x +1) is a factor of x³ + 5x² + kx –12

Find K

Solution:

If (x +1) is a factor then x = -1 is a root.

(-1)³ + 5(-1)² + k(-1) –12 = 0

                -1 + 5 – k – 12 = 0

                      -1 + 5 – 12 = k

                                    -8 = k

 

 

 

Example 2:

(x + 3) is a factor x³ + 5x² + 7x +3

Find the other two factors.

Solution:

 

 

 

 

No remainders.

(x² + 2x + 1) = (x +1) (x + 1)

Factors of f(x) = (x + 3) (x +1) (x + 1)

 

 

 

Example 3:

F(x) = x³ + 5x² + 2x – 8 and f(-4) = 0

Find the factors.

Solution:

If f(-4) = 0 we can derive that (x + 4) is a factor.

 

 

Divides exactly.

(x² + x – 2) = (x + 2) (x – 1)

Factors of f(x) = (x + 4) (x + 2) (x – 1)