Question 1144164
<br>
Prediction: {{{(x^5+2x^4+2x^3+2x^2+2x+1)/(x+1) = x^4+x^3+x^2+x+1}}}<br>
Explanation....<br>
Look at the familiar product when (x+1) is multiplied by (x+1) and like terms are combined in the product:<br><pre>
(x+1)(x+1) = x^2 + x
           +       x + 1
           --------------
             x^2 +2x + 1</pre>
Now look what we get when (x^2+x+1) is multiplied by (x+1):<br><pre>
(x^2+x+1)(x+1) = x^3 + x^2 + x 
               +       x^2 + x + 1
                -------------------
                 x^3 +2x^2 +2x + 1</pre>
You should be able to see that, as more powers of x are added to the first factor, the pattern of the product will continue.  So in particular<br><pre>
x^4+x^3+x^2+x+1)(x+1) = x^5 + x^4 + x^3 + x^2 + x
                      +       x^4 + x^3 + x^2 + x + 1
                      -------------------------------
                        x^5 +2x^4 +2x^3 +2x^2 +2x + 1</pre>
And so<br>
{{{(x^5+2x^4+2x^3+2x^2+2x+1)/(x+1) = x^4+x^3+x^2+x+1}}}<br>