Question

5. Which polynomial is equal to (x^5+ 1) divided by (x + 1)?

AXA - X3 .x² - x + 1.
B X - X² + x² - x + 1
C x4 + x3 -- x2 + x + 1
D x + x3 + x² + x 1

Answer 1

B
`x^4-x^3+x^2-x+1`

Explanation:

Given,

Dividend =
`(x^5+1)`

Divisor =
`(x+1)`

Now According to the rule of Division.

Step 1: At first dividend is
`(x^5+1)` and Divisor is
`(x+1)` when it is divided for the first time the quotient will be
`x^4` and remainder will be
`-x^4+1`

Step: 2 Now the remainder of step 1 will be new dividend which is
`-x^4+1` and Divisor is
`(x+1)` so when it is divided the quotient will be
`x^4-x^3` and remainder will be
`x^3+1`

Step: 3 Now the remainder of step 2 will be new dividend which is
`x^3+1` and Divisor is
`(x+1)` when it is divided the quotient will be
`x^4-x^3+x^2` and remainder will be
`-x^2+1`

Step: 4 Now the remainder of step 3 will be new dividend which is
`-x^2+1` and Divisor is
`(x+1)` when it is divided the quotient will be
`x^4-x^3+x^2-x` and remainder will be
`x+1`

Step: 5 Now the remainder of step 4 will be new dividend which is
`x+1` and Divisor is
`(x+1)` when it is divided the quotient will be
`x^4-x^3+x^2-x+1` and remainder will be 0.

Hence When the polynomial
`(x^5+1)` is divided by
`(x+1)` the answer or quotient will be equal to
`x^4-x^3+x^2-x+1` and remainder will be 0.