Question 1172954
<br>
(previous response replaced -- contained arithmetic errors....)<br>
By the rational roots theorem, the possible rational roots are<br>
(plus or minus) {1, 2, 4, 5, 8, 10, 20, 40; 1/2, 5/2}<br>
You can test each rational root using synthetic division; but that is very tedious.<br>
Finding the roots using a graphing calculator is fast and easy.  The factorization is<br>
{{{(x-4)(2x-1)(x^2-2x-10)}}}<br>
The quadratic formula gives the roots of the quadratic factor as<br>
{{{1+sqrt(11)}}} and {{{1-sqrt(11)}}}<br>
So if you really need to write the polynomial as a product of linear factors, it is<br>
{{{(x-4)(2x-1)(x-(1+sqrt(11)))(x-(1-sqrt(11)))}}}<br>