Question

using remainder theorem factorise 2x³+x²-2x-1

Answer 1

2x³+x²-2x-1= x²( 2x+1) -1(2x+1) = (x²-1) (2x+1) = (x+1) (x-1) (2x+1)

Answer 2

Given :   2x³  + x² - 2x - 1

To Find : Using the remainder theorem, factorise

Solution:

2x³  + x² - 2x - 1

polynomial p(x) divided by (x - a)

Then p(a) is the remainder

if p(a) = 0 then (x - a) is  a factor

p(x)  = 2x³  + x² - 2x - 1

p(1) = 2(1) + 1² - 2(1) - 1  = 0

Hence x - 1 is one of the factor

p(-1)  = 2(-1) + (-1)² - 2(-1) - 1  = - 2 + 1 + 2 - 1 = 0

Hence ( x- (-1)) = x + 1 is also a factor

2x³  + x² - 2x - 1   is divisible by (x + 1)(x - 1)

(x + 1)(x - 1) = x² - 1

              2x + 1

x² - 1   _|  2x³  + x² - 2x - 1   |_

               2x³         -2x

             _______________

                         x²        - 1

                         x²        - 1

             _______________

                              0

2x³  + x² - 2x - 1  =  (x + 1)(x - 1)(2x + 1)

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