Question

find the remainder x⁴ +x³ -2x²+x+1is divided by x-1 ​

Answer 1

Step 1: Dividend = x⁵, Divisor = x+1. Q1 = x⁴ and R1 = -x⁴

Step 2: Dividend =-x⁴, Divisor = x+1. Q2 = -x³ and R2 = +x³

Step 3: Dividend =+x³, Divisor= x+1. Q3 = +x²and R3 =-x²

Step 4: Dividend =-x², Divisor = x+1. Q4 = -x and R4 =+x

Step 5: Dividend =+x , Divisor = x+1. Q5 = +1 and R5 =-1

As R5 is less than the divisor x+1, we terminate the division here.

∴ Q = Q1 + Q2 + Q3 + Q4 + Q5 = x⁴ - x³ + x² - x + 1

and R = -1

Step-by-step explanation:

E = Q x D + R .

In our case, E = x⁵, D = x + 1, Q = x⁴ - x³ + x² - x + 1, R = -1

∴ Q x D + R = (x⁴ - x³ + x² - x + 1) (x+1) + (-1)

Multiplying term by term

=x⁵ - x⁴ + x³ - x² + x + x⁴ - x³ + x² - x + 1 - 1

= x⁵ (Cancelling equal terms)

= E (Proved)

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Answer 2

x-1 = 0 → x = 1
P(1) = 1+1-2+1+1
Remainder = 2