Question

if p(x)=2x^4+3x^3-3x^2-2x+5 is divided by 2x^2+3x-1 then the remainder is x-a.find a.

Answer 1

P(x) = 2x⁴ + 3x³ + 3x² + 2x + 5

Rearranging and rewriting the above equation so as to get the divisor 2x² + 3x + 1,

P(x) = 2x⁴ + 3x³ + x² + 2x² + 2x +1+4

= P(x) = x²(2x² + 3x + 1) + 2x² + 2x + 1 + 4

Again for making the divisor adding and subtracting x in above equation

P(x) = x²(2x² + 3x + 1) + 2x² + 2x + x + 1 - x + 4

= P(x) = x²(2x² + 3x + 1) + (2x² + 3x + 1) - x + 4

Now as the first two terms of P(x) contain the divisor they are divisible by it leaving remainder zero.Hence the remainder is equal to the last term .

= Remainder = -x + 4

Thus a = 4

Answer 2

P(x) = 2x⁴ + 3x³ + 3x² + 2x + 5

Rearranging and rewriting the above equation so as to get the divisor 2x² + 3x + 1,

P(x) = 2x⁴ + 3x³ + x² + 2x² + 2x +1+4

= P(x) = x�(2x² + 3x + 1) + 2x² + 2x + 1 + 4

Again for making the divisor adding and subtracting x in above equation

P(x) = x²(2x² + 3x + 1) + 2x² + 2x + x + 1 - x + 4

= P(x) = x²(2x² + 3x + 1) + (2x² + 3x + 1) - x + 4

Now as the first two terms of P(x) contain the divisor they are divisible by it leaving remainder zero.Hence the remainder is equal to the last term .

= Remainder = -x + 4

Thus a = 4

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