Divide x^4+1 by x+1 and verify division algorithm
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Answer:
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,
Hence, Quotient=q(x) = x³ - x² + x - 1
Hence, Quotient=q(x) = x³ - x² + x - 1Remainder=r(x) = 0
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