polynomials (1 Viewer)

[wgy182]

Hey guys,
I don't know how to use latex so sorry if its abit difficult to read.

Question: Given P(z)= z^4 +az^3+bz-1,find the value of a and b if I and 1+√2 are zeros of P(z).

I got up to this,
P(i) -ai+bi=-2
P(1+√2) 7a+5√2a+b+√2b=-16-12√2

How do I solve it simultaneously? Or is there another way in doing this question?


Thanks in advance

 

[QZP]

use coefficient-root relationships (& complex conjugate root thereom)

 

[wgy182]

Can you show me how?

 

[Troller]

If the polynomial's coefficients are real then -i is a root.

 

[dunjaaa]

So if P(z) is a real polynomial, then complex roots occur in conjugate pairs. Thus z=-i is also a root.
Let the final root of P(z) be (alpha)
Product of the Roots = (i)(-i)(1+√2)(alpha)= -1
(alpha)=1-√2
Therefore the roots are z=i, -i, 1+√2, 1-√2
Sum of roots = -a
a = -2
Sum of roots (3 at a time) = -b
(i)(-i)(1+√2)+(i)(-i)(1-√2)+(-i)(1+√2)(1-√2)+(i)(1+√2)(1-√2) = -b
b = -2

 

[wgy182]

Thanks guys!!!