Remainder Theorem

Show that (x - 1) is a factor of x³ – 7x ² + 14x – 8. Hence completely factorize

                                                           (ICSE 2007)

Answer:

Let: F(x) = x³ – 7x ² + 14x – 8

Since (x - 1) is a factor of F(x), therefore, according to Remainder theorem,

we have, F (1) = 0

Now, F (1) = (1)³ - 7(1)² + 14(1) – 8

Or, F(1) =1 – 7 +14 - 8 = 0

So, (x – 1) is a factor.

Now we will divide (x³ – 7x ² + 14x – 8)/(x – 1)

(x - 1) // x³ – 7x ² + 14x – 8 //  x ² - 6x + 8

             x³ - x ² 

               - 6 x ² + 14x

              - 6 x ² +  6x

                            8x – 8

                           8x – 8

                           0     0  

(x ² - 6x +8) is the other factor of F(x). We will further simplify, thus:

x ² - 6x +8 = x ² - 4x - 2x + 8 = x(x – 4) – 2(x – 4) = (x – 2)(x – 4)

So, x³ – 7x ² + 14x – 8  = (x – 1 )(x – 2)(x – 4)