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.