given the xeros 0, 2, 1 + i, 1 - i, write the polynomial

[Stranger_1973]

Given the xeros 0, 2, 1 + i, 1 - i, write the polynomial

I have no idea

[stapel_eliz]

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!

[Stranger_1973]

how did you get x - 1 - i?

[stapel_eliz]

how did you get x - 1 - i?

By working backwards from the zero, and by using parentheses to make my meaning clear (to myself).

If x = 2 is a zero, then (subtracting) x - 2 = 0 was what they solved, so x - 2 was a factor.

If x = 1 + i is a zero, then (subtracting and using grouping symbols) x - (1 + i) = 0 was what they solved, so x - (1 + i) = x - 1 - i was a factor.

Use the same method to find the last factor. Then multiply everything together. It will be simplest if you multiply the two complex-valued factors first!

[Stranger_1973]

so the other zero is x = 1 - i, so x - (1 - i) = x - 1 + i? how do you multiply it?

[stapel_eliz]

so the other zero is x = 1 - i, so x - (1 - i) = x - 1 + i?

Yes!

how do you multiply it?

I'd work vertically, and probably keep the parentheses:

vertical set-up:
                    x - (1 + i)
                    x - (1 - i)
-------------------------------
    - (1 - i)x + (1 + i)(1 - i)
x^2 - (1 + i)x
-------------------------------

Multiplying the conjugates is easy: that's just the reverse of factoring a difference of squares. The rest simplifies nicely, and you can add down to get the quadratic below the second "equals" bar.


Edit: Inserting missing "minus" sign in first term of third line of multiplication. Thank you, DAiv!

[Stranger_1973]

i got it
thanks