Question

Without actual division, prove that 2x⁴ – 5x³ + 2x² – x + 2 is divisible by x² – 3x + 2. [Hint: Factorise x² – 3x + 2]

Answer 1

Let p(x) = 2x⁴ – 5x³ + 2x² – x + 2 firstly, factorise x² – 3x + 2.

Now, x² – 3x + 2 = x² – 2x – x + 2  ...[By splitting middle term]

= x(x – 2) – 1(x – 2) = (x – 1)(x – 2)

Hence, 0 of x² – 3x + 2 are 1 and 2.

We have to prove that, 2x⁴ – 5x³ + 2x² – x + 2 is divisible by x² – 3x + 2 i.e., to prove that p(1) = 0 and p(2) = 0

Now, p(1) = 2(1)⁴ – 5(1)³ + 2(1)² – 1 + 2

= 2 – 5 + 2 – 1 + 2

= 6 – 6

= 0

And p(2) = 2(2)⁴ – 5(2)³ + 2(2)² – 2 + 2

= 2x¹⁶ – 5x⁸ + 2x⁴ + 0

= 32 – 40 + 8

= 40 – 40

= 0

Hence, p(x) is divisible by x² – 3x + 2.