Remainder Theorem

Use the Remainder Theorem to factorize the following expression:

 2x³ +x² - 13x + 6.               (ICSE 2010)

Answer:

Substitute 1,2 and 3 to check if F(x) =  2x³ +x² - 13x + 6 = 0

We find that for x = 2, F(2) = 0

So, (x -2) is a factor.

The other factor is of the form ax² + bx + c, because F(x) is a trinomial.

Therefore (x-2)( ax² + bx + c) = 2x³ +x² - 13x + 6

Or, ax³ + (b -2a)x² – (c -2b)x – 2c =2x³ +x² - 13x + 6

Comparing the coefficients of the individual powers of x on LHS and RHS,

We have a = 2,

(b – 2a) = 1

Or, b = 5

And -2c = 6

Or, c = -3

So, ax² + bx + c = 2x² + 5x – 3 = 2x² + 6x – x – 3= 2x(x + 3) – 1(x + 3) = (2x – 1)(x + 3)

Therefore, 2x³ +x² - 13x + 6 = (x-2)(2x – 1)(x + 3)  answer