Tuesday, July 23, 2013

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)

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