Question

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

Answer 1

Answer:

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Answer 2

Answer:

Let f(x) = 2x⁴ - 5x³ + 2x² - x + 2 and g(x) = x² - 3x + 2

We have,

g(x) = x² - 3x + 2 = (x-1) (x-2).

Clearly, ( x - 1 ) and ( x + 2 ) are factors of g (x).

• In order to prove that f(x) is exactly divisible by g(x), It is sufficient to prove that f(x) is divisible by (x-1) and (x-2). For this we will show that (x-1) and (x-2) are factors of g(x).

We have ,

f(1) = 2(1)⁴ - 5(1)³ + 2(1)² - 1 + 2

=> 2 - 5 + 2 - 1 + 2 = 0.

and ,

f(2) = 2(2)⁴ - 5(2)³ + 2(2)² - 2 + 2

=> 32 - 40 + 8 - 2 + 2 = 0

•°• ( x - 1 ) and ( x + 2 ) are factors of g(x).

=> g(x) = ( x - 1 ) ( x - 2 ) is a factor of p(x).

Hence, p(x) is exactly divisible by g(x).