Question

using long division method, show that the polynomial p(x)=x³+1 is divisible by q(x)=X+1. verify your result using factor theorem

Answer 1

Given p(x) = x^3 + 1, g(x) = x + 1.

Long Division Method:

x + 1) x^3         +        1 ( x^2 - x + 1.

        x^3 + x^2

        -------------------------

                   -x^2       +  1

                   -x^2 - x

          ---------------------------

                               x +  1

                               x +  1

          ------------------------------

                                    0.

Factor theorem:

Given p(x) = x^3 + 1 and q(x) = x + 1.

We know that by factor theorem, if x + 1 is a factor, then x = -1 is a root.

Substitute x = -1 in p(x), we get

⇒ f(-1) = (-1)^3 + 1

          = -1 + 1

          = 0.

The remainder is 0, so the factor theorem says that:

x + 1 is a factor of x^3 + 1.

Hope this helps!

Answer 2

Look  in attachment ,………