polynomials (1 Viewer)

[AFGHAN22]

please help with this question guys!!!

Given P(z) = z^4 + az^3 + bz -1 , find the values of a and b if i and 1 + sqrt 2 are zeros of p(z).
answer: 4 , -4

thanks in advance

 

[Mountain.Dew]

please help with this question guys!!!

Given P(z) = z^4 + az^3 + bz -1 , find the values of a and b if i and 1 + sqrt 2 are zeros of p(z).
answer: 4 , -4

thanks in advance

heres my 2 cents for this question:

okay, must realise that u have two roots given to you. SPECIAL ROOTS. roots that have CONJUGATES. there has to be conjugates because the coefficients of P(z) are real. this is a property of polynomials that all 4U and 3U maths students must etch into their minds. If there are only real coefficients in polynomial, there must be conjugate non-real roots and surded conjugate roots.

SO, since one of the roots is i, then another root of P(z) is -i.
same with 1+ sqrt 2 --> the last unknown root is 1 - sqrt 2.

SO, we have the roots i, -i, 1-sqrt2, 1 + sqrt 2.

then, there are two ways to find a and b...

1st method: do sum of roots to find a, and do sum of double roots to find b.
2nd method: expand (x-i)(x+i)(x-1-sqrt2)(x-1+sqrt2), and equate coefficients with their respective terms of x^3 and x^2 to get a and b.

 

[AFGHAN22]

thanks mate, greatly appreciated

 

[Mountain.Dew]

*tosses another 2 cent coin*

not a problem.