Question

Without actual division, prove that 2x4 - 5x3 +2x2 –x +2 is divisible by x2 -3x +2.

Answer 1

Answer:

= x^2 - 3x + 2

= x^2 - ( 2 + 1 )x + 2

= x^2 - 2x - x + 2

= x( x - 2 ) - ( x - 2 )

= ( x - 2 )( x - 1 )

If 2x^4 - 5x^3 + 2x^2 - x + 2 is divisible by x^2 - 3x + 2, then the 2x^4 - 5x^3 + 2x^2 - x + 2 must be divisible by the factors of x^2 - 3x + 2 as well.

Henceforth, 2x^4 - 5x^3 + 2x^2 - x + 2 should be divisible by x - 2 and x - 1.

Using factor theorem-

For x = 2, polynomial should be 0

= > 2(2)^4 - 5(2)^3 + 2(2)^2 - (2) + 2

= > 2(16) - 5(8) + 2(4) - 2 + 2

= > 32 - 40 + 8

= > 0

Hence x - 2 is a factor of the given polynomial of degree 4

For x = 1, polynomial should be 0

= > 2(1)^4 - 5(1)^3 + 2(1)^2 - 1 + 2

= > 2 - 5 + 2 - 1 + 2

= > 6 - 6

= > 0

Hence x - 1 is a factor of the polynomial of degree 4

And therefore, 2x^4 - 5x^2 + 2x^2 - x + 2 is divisible by x^2 - 3x + 2.

Hence Proved.