Question

Using long divisible method ,show that the polynomial p(x)=x³+1 is divisible by q(x)=x+1. Verify your result using factor thorem

Answer 1

Answer:

Long division

        x^2 - x + 1

        ____________

x+1  [ x^3             + 1

        [ x^3 + x^2

          ----------------

                   -x^2         + 1

                   -x^2 - x

                   ------------

                           x   + 1

                           x  +   1

                           ----------

                              0   

so, the answer is:

x² - x + 1

Using factor theorm:

a³ + b³ = (a + b)(a² – ab + b²)

So, x³ + 1 = x³ + 1³ = (x+1)(x² - 2x + 1)

and (x³ + 1) / (x+ 1) = (x+1)(x² - 2x + 1) / (x + 1)

The (x + 1) brackets cancel out and we are left with

x² - 2x + 1