Find the quotient and remainder when (x⁴+1) . . .

Question : Find the quotient and remainder when (x⁴+1) is divided by (x-1) by actual division. Also, verify the remainder by remainder theorem.

Doubt by Saksham

Solution :

Let p(x) = x⁴+1 g(x) = x-1

Now Dividing p(x) by g(x) by long division method

_______________

x-1) x⁴+1 (x³+x²+x+1

x⁴ - x³

- +

------------ x³+1

x³+ - x² - + ------------ x²+1

x² - x - + ------------- x + 1 x - 1 - + -------------- 2 --------------

∴ quotient, q(x) = x³+x²+x+1 & remainder r(x) = 2

Now, Verification by using Remainder Theorem

p(x) = x⁴+1 g(x) = x-1

g(x) = 0 x-1 = 0 x=1

p(x) = x⁴+1 p(1) =(1)⁴+1

p(1) = 1+1 p(1) = 2 ∴ Remainder, r(x) = 2

Hence remainder is verified.