Question

answer this please.....

Answer 1

P(x) = x^4 + x^3 +x^2 +x+ 1

g(x) = x^2 +2x +1

g(x) = x^2 +x +x +1

g(x) = x(x+1) + 1(x+1)

g(x) = (x+1)^2

(x+1)^2 = 0

=> x +1 = 0

=> x = - 1

By Remainder theoram,

P(-1) = (-1)^4 + (-1)^3 +(-1)^2 + (-1)+1

= 1 - 1 +1 - 1 +1

= 3-2

= 1

Remainder = 1

So, g(x) is not a factor of P (x)

Answer 2

Given f(x) = x^4 + x^3 + x^2 + x + 1.

Given g(x) = x^2 + 2x + 1.

                  = x^2 + 1x + 1x + 1

                  = (x + 1) + 1(x + 1)

                  = (x + 1)(x + 1).


Now we have to show that (x + 1)(x + 1) are factors of f(x).

Using factor theorem,

When f(x) is divided by x + 1, the remainder is given by

f(-1) = (-1)^4 + (-1)^3 + (-1)^2 + (-1) + 1

      = 1 - 1 + 1 - 1 + 1

      = 1.


Therefore x^2 + 2x + 1 is a factor of f(x).

Hence f(x) is divisible by g(x).


Hope this helps!