Question 1126759
<br>
{{{x^4 +3x^3-8x^2 +3x+1=0}}}<br>
Divide the whole equation by x^2.  (We can do this because we can see by inspection that x=0 is not a root.)<br>
{{{x^2+3x-8+3/x+1/x^2 = 0}}}<br>
{{{x^2+2+1/x^2}}} is (x+1/x) squared.  Use this to rewrite the expression on the left in the equation as a polynomial with "x+1/x" as the variable; then factor.<br>
{{{(x^2+2+1/x^2)+3x+3/x-10 = 0}}}  [group terms as appropriate; note the "-8" is now represented as "2...-10"]<br>
{{{(x+1/x)^2+3(x+1/x)-10 = 0}}}  [this is now a polynomial with "x+1/x" as the variable.  Substitute u = x+1/x if it helps you see what is being done]<br>
{{{((x+1/x)+5)((x+1/x)-2) = 0}}}<br>
Now multiply each factor by x (to "un-do" the first step above where we divided the whole equation by x^2)<br>
{{{(x^2+5x+1)(x^2-2x+1) = 0}}}<br>
{{{x^2+5x+1 = 0}}} or {{{x^2-2x+1 = 0}}}<br>
{{{x = (-5 +- sqrt(21))/2}}} or {{{x = 1}}}