Find an equation given the roots of 3, 2 + i and 2 - i.
This is actually pretty easy. Remember: Roots are what causes and equation to be zero. So, in this case, we multiply the following (the given roots are in bold red) in order to find the equation:
y = (x - 3)(x - (2 + i))(x - (2 - i))
Now, just evaluate:
y = (x - 3)(x - 2 - i)(x - 2 + i)
Think FOIL method for one part of it:
y = (x - 3)(x2 - 2x + ix - 2(x - 2 + i) - i(x - 2 + i))
y = (x - 3)(x2 - 2x + ix - 2x + 4 - 2i - ix + 2i - i2)
y = (x - 3)(x2 - 4x + 4 -(-1))
y = (x - 3)(x2 - 4x + 5)
y = x(x2 - 4x + 5) - 3(x2 - 4x + 5)
y = x3 - 4x2 + 5x - 3x2 + 12x - 15
y = x3 - 7x2 + 17x - 15
Take notice, that I go step-by-step as if writing a computer program. This is optimal for solving math problems. Take your time! It prevents mistakes such as miscalulations and reversing signs. And if you are given more roots, there will be more "computing" and more uses of the FOIL method.
Notice that you had 3 roots and the highest power of the equation is also 3, as in x3. If you had roots 2i, -2i, 3i, -3i and 1, then that is 5 roots. The equation would start with x5. And if you use the approach above you get the following:
y = x5 - x4 + 13x3 - 13x2 + 36x - 36
Go ahead and give it a try and see if you get the same result.