Question

divide x 4 + 1 by X + 1 and verify division of algorithm​

Answer 1

Solution :-

Given : f(x) = x⁴ + 1

g(x) = x + 1

Let q(x) = ax³ + bx² + cx + d

r(x) = e

Applying division algorithm,

∴ f(x) = g(x).q(x) + r(x)

=> x⁴ + 1 = (x + 1)(ax³ + bx² + cx + d) + e

=> x⁴ + 1 = ax⁴ + bx³ + cx² + dx + ax³ + bx² + cx + d + e

=> x⁴ + 1 = ax⁴ + (a + b)x³ + (b + c)x² + (c + d)x + (d + e)

Here,

Coefficient of x⁴ = a = 1

Coefficient of x³ = a + b = 0

=> 1 + b = 0

=> b = - 1

Coefficient of x² = b + c = 0

=> - 1 + c = 0

=> c = 1

Coefficient of x = c + d = 0

=> 1 + d = 0

=> d = - 1

Constant = d + e = 1

=> - 1 + e = 1

=> e = 0

Hence,

Quotient = q(x) = x³ - x² + x - 1

Remainder = r(x) = 0