Remainder Theorem

1. Given that (x - 2) is a factor of 2x³ – x ² – px – 2. Determine the values of “p.         

                                                                                      (ICSE 2008)

Answer:

Let: F(x) = 2x³ – x ² – px – 2

Since (x - 2) is a factor of F(x), therefore, according to Remainder theorem, we have, F (2) = 0

Now, F (2) = 2(2)³ - (2)² - p(2) – 2 = 0

Or, 16 - 4 -2p – 2 = 0

Or, 2p = 10 

Or, p = 5  Answer

2.     Given that (x - 2) is a factor of 2x³ – x ² – px – 2. With the              value of p factorize the expression completely.         

                                                                                   (ICSE 2008)

Answer:

Since (x - 2) is a factor of 2x³ – x ² – 5x – 2.

The other factor has to be a trinomial of the form ax² + bx + c

Therefore (x-2)( ax² + bx + c) = 2x³ – x ² – 5x – 2.

Or, ax³ + (b -2a)x² – (c -2b)x – 2c = 2x³ – x ² – 5x – 2

Comparing the coefficients of individual power of x,  

We have a = 2,

(b – 2a) = -1

Or, b = 3

And -2c = -2

Or, c = 1

So, ax² + bx + c = 2x² + 3x +1 = 2x² + 2x + x +1= 2x(x + 1) + 1(x + 1) = (2x + 1)(x + 1)

Therefore, 2x³ – x ² – 5x – 2 = (x-2)(2x + 1)(x + 1)