Stranger_1973 wrote:Given the xeros 0, 2, 1 + i, 1 - i, write the polynomial
When you solved polynomials (starting with solving quadratic equations), you factored, set the factors equal to zero, and solved the resulting linear equations. When you couldn't factor the quadratic down to linear factors (like "3x + 4"), you solved the quadratic factor by applying
the Quadratic Formula.
In general, if you had factors (x - a) and (x - b), then you set the factors equal to zero and solved: x - a = 0 and x - b = 0, so x = a and x = b.
Now you're being asked to
work backwards: Given the zeroes x = a and x = b, you know that they must have solved x - a = 0 and x - b = 0, so the factors must have been (x - a) and (x - b). You then find the original quadratic by multiplying these factors.
In your case, the original polynomial obviously required the Quadratic Formula to find two of the zeroes: that's where the imaginaries came from. But the process is exactly the same. If 0 is a zero, then x = 0 was an equation, so x was a factor. If 2 is a zero, then x = 2 must have been an equation, so they must have solved x - 2 = 0, so... what was a factor? If 1 + i is a zero, then x = 1 + i must have been a solution of the Quadratic Formula, so x - 1 - i must have been a factor. If 1 - i is a zero, then... what must have been the other factor?
Multiply it all out to find the polynomial they're looking for.
Have fun!