Question

Using long division show that polynomial p(x) = x^3 + 1 is divisible by q(x) = x + 1 .Verify your result using factor theorem.

Answer 1

Answer:

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