[RESOLVED] Polynomial of degree 4.

[eranga262154]

Anyone can help me on this.

A polynomial P(z) of degree 4 with real coefficient has roots z=i, z=i+1 and satisfies P(0) = 2. Find P(z).

I start to work on given conditions, but go through it difficult find more equations. Polynomial standard format I used is,

P(z) = z⁴ + az³ + bz² + cz + d

So d = 2 from third condition. To find all other four coefficient a,b and c, I find only two equations. Can't go further.

[zaza]

Don't forget that the z⁴ term has a coefficient as well.

As you say, P(0) = 2 so that gives you the constant. Then by substituting in for z = i and z = i+1, and bearing in mind that the imaginary parts sum to 0 and the real parts sum to 0, you have 4 equations. Real part of first = 0, Im part of first = 0, etc

zaza

[eranga262154]

Don't forget that the z⁴ term has a coefficient as well.

Correct. But I neglect is for my clarification. Finally I can find that coefficient. That is, if unknown terms a,b,c,d derived as a fractions, GCD is the coefficient of z⁴

As you say, P(0) = 2 so that gives you the constant. Then by substituting in for z = i and z = i+1, and bearing in mind that the imaginary parts sum to 0 and the real parts sum to 0, you have 4 equations. Real part of first = 0, Im part of first = 0, etc

Thanks, I'll try it.

[zaza]

Correct. But I neglect is for my clarification. Finally I can find that coefficient. That is, if unknown terms a,b,c,d derived as a fractions, GCD is the coefficient of z⁴

Don't neglect the z⁴ coefficient and try to find it at the end. It will make your life a lot easier if you include it at the start.

[eranga262154]

Depend on the question it can be easy or difficult. Mostly I feel it easy. Anyway thanks.

When I'm going to this, coefficient on z⁴ make the evaluation too difficult to me.