# Thread: Help forming polynomial of least degree

1. ## Help forming polynomial of least degree

5. b) Find a real polynomial of least degree having 1+i, 2, -i as roots.

My work:

1.
x=1+i
x=2
x= -i

2.
[(x-1)-i]
[(x-2)]
[(x)+i]

3.
Secondly I tried putting them together and got:

(x-1)(x) + (x-1)i - (x)i - i^2

(x-1)(x) + (x-1)i - (x)i + 1

(x-1)(x) - i + 1

2. Originally Posted by thekrown
5. b) Find a real polynomial of least degree having 1+i, 2, -i as roots.

My work:

1.
x=1+i
x=2
x= -i

2.
[(x-1)-i]
[(x-2)]
[(x)+i]

3.
Secondly I tried putting them together and got:

(x-1)(x) + (x-1)i - (x)i - i^2

(x-1)(x) + (x-1)i - (x)i + 1

(x-1)(x) - i + 1

The roots of a polynomial whose coefficients are all real always occur in conjugate pairs ....

3. Okay so we would have...

[(x-1)-i]
[(x-1)+i]

[x-2]

[(x)+i]
[(x)-i]

and then I would form the poly with these?

4. Originally Posted by thekrown
Okay so we would have...

[(x-1)-i]
[(x-1)+i]

[x-2]

[(x)+i]
[(x)-i]

and then I would form the poly with these?
Take the product of these factors.

Note that $\displaystyle ([x - 1] - i)([x - 1] + i) = (x - 1)^2 + 1 = x^2 - 2x + 2$ etc.

5. I end up with [(x-1)^2+1] * [x-2] * [x^2+1]

Is there a technique to shorten the multiplication process?

6. Originally Posted by thekrown
I end up with [(x-1)^2+1] * [x-2] * [x^2+1]

Is there a technique to shorten the multiplication process?
Why expand? Does the question ask you to expand? This is a correct answer. Unless you have good reason to do otherwise, leave it at that.

7. The question is exactly as I typed above, so no they do not ask to expand. Good thing too.

I always thought polynomials to be expanded as you say.

Well thank you, you helped me a lot.