Question 29538
<pre><b>how do you determine an equation from the zeros 
for example 
-3/4, 2/5 
-4+2(square root)3
-4-2(square root)3 

To find a polynomial equation whose zeros are r1, r2, r3, r<sub>4</sub>, form the 
following equation, multiply it out and simplify it.

(x - r<sub>1</sub>)(x - r<sub>2</sub>)(x - r<sub>3</sub>)(x - r<sub>4</sub>) = 0
                            _          _
For r<sub>1</sub>=-3/4, r<sub>2</sub>=2/3, r<sub>3</sub>=-4+2<font face = "symbol">Ö</font>3, r<sub>4</sub>=-4-2<font face = "symbol">Ö</font>3)

Such an equation would be:
                             _             _ 
(x + 3/4)(x - 2/5)[x - (-4+2<font face = "symbol">Ö</font>3)][x - (-4-2<font face = "symbol">Ö</font>3)] = 0

Multiply that out -- a very big job -- then you'll get

x<sup>4</sup> + (167/20)x<sup>3</sup> + (13/2)x<sup>2</sup> - x - 6/5 = 0

Clearing of fractions

20x<sup>4</sup> + 167x<sup>3</sup> + 130x<sup>2</sup> - 20x - 24 = 0

Edwin
AnlytcPhil@aol.com</pre>