Question 196689
Find the roots of the polynomial equation.

{{{x^4 - 5x^3 + 11x^2 - 25x + 30 = 0}}}; 2,3
<pre><font size = 4 color = "indigo"><b>
Since you are given that 2 is a root, we
do this synthetic division

2 | 1   -5   11  -25  30
  |      2   -6   10 -30 
   ---------------------
    1   -3    5  -15   0

So now we have factored the original polynomial as

{{{(x-2)(x^3-3x^2+5x-15)=0}}}

Now we factor the cubic polynomial in 
the 2nd parentheses, using the fact that 3 is
a root of the original, so it must be a 
root of the cubic polynomial in the 2nd
parentheses:

3 | 1 -3  5 -15
  |    3  0  15 
   ---------------
    1  0  5   0

So now we have factored the original polynomial as

{{{(x-2)(x-3)(x^2+0x+5)=0}}}

or

{{{(x-2)(x-3)(x^2+5)=0}}}

Setting the first two parentheses = 0 gives
the given two roots, 2 and 3.  To find the
other roots, we set the third factor = 0

{{{x^2+5=0}}}

Add -5 to both sides

{{{x^2=-5}}}

Take the square roots of both sides:

{{{sqrt(x^2)=" "+-sqrt(-5)}}}

{{{x = " "+-sqrt(5)i}}}

So the roots are 2, 3, {{{sqrt(5)i}}}, and {{{-sqrt(5)i}}}

Edwin</pre>