Question 717927
Either there is a mistake in
{{{f(x)=x^8+8x^7-28x^6-56x^5+70x^4+56x^3-28x^2-8x+1}}}
or I don't believe it is possible to find the exact real roots for this polynomial.<br>
If what you posted is correct, then there are no rational roots. The only possible rational roots are 1 and -1 and neither of them work out to be roots. So the only real roots of this function would be irrational. The best one could do, I believe, is to use a graphing calculator:<ol><li>Graph {{{y=x^8+8x^7-28x^6-56x^5+70x^4+56x^3-28x^2-8x+1}}}</li><li>Then use the trace function to find decimal approximations for the x-coordinates of all the places where the graph intersects the x-axis. (If the graph never intersects the x-axis then there are no real roots.) I'll show you a graph at then end.</li></ol><br>
I suspect that the function is supposed to be:
{{{f(x)=x^8-8x^7+28x^6-56x^5+70x^4-56x^3+28x^2-8x+1}}}
Again, the only possible rational roots are 1 and -1. But this time 1 is actually a root. Here's the synthetic division:
<pre>
1  |   1   -8   28   -56   70   -56   28   -8   1
----        1   -7    21  -35    35  -21    7  -1
      --------------------------------------------
       1   -7   21   -35   35   -21    7   -1   0  <== a root!
Trying 1 again:
1  |   1   -7   21   -35   35   -21    7   -1 
----        1   -6    15  -20    15   -6    1    
      ----------------------------------------
       1   -6   15   -20   15    -6    1    0  <== a root!
Trying 1 again:
1  |   1   -6   15   -20   15    -6    1
----        1   -5    10  -10     5    1    
      -----------------------------------
       1   -5   10   -10    5    -1    0  <== a root!
Trying 1 again:
1  |   1   -5   10   -10    5    -1 
----        1   -4     6   -4     1        
      ------------------------------
       1   -4    6    -4    1     0  <== a root!
</pre>I think by now you can see where this is going. 1 is a root 8 times. In other words it is a root of multiplicity 8. And f(x) in factored form is:
{{{f(x) = (x-1)^8}}}<br>
FWIW, The following is the graph of the function you posted. It shows 5 points where the graph intersects the x-axis. But there are more such points outside of what we see on this graph. There will be 1, 2 or 3 additional points where the graph intersects the x-axis. We know there is at least 1 because an 8th degree polynomial which has a positive leading coefficient will go toward infinity for large positive and negative x's. So that steep "dive" we see at about x = 1.2 has to eventually come back up and cross the x-axis somewhere out to the right.
{{{graph(800, 800, -5, 15, -10, 10, x^8+8x^7-28x^6-56x^5+70x^4+56x^3-28x^2-8x+1)}}}