QUESTIONS - POLYNOMIALS COMPLEX ANALYSIS

QUESTIONS - ANSWERS -SUNNY- POLYNOMIALS - COMPLEX ANALYSIS

FIND IF (3-5i) IS A FACTOR OF
4X^4-28X^3+149X^2-70X-374.

USING REMAINDER THEOREM TO CHECK COMPLEX ROOTS IS OFTEN CUMBERSOME PARTICULARLY WITH LARGE COEFFICIENTS AS THIS AND 4 TH. DEGREE POLYNOMIAL
HENCE IT IS BETTER TO DO IT THE FOLLOWING WAY.
IF ….3-5i IS A FACTOR,THEN ITS CONJUGATE 3+5i SHOULD BE A FACTOR AS THE POLY NOMIAL HAS REAL COEFFICIENTS.
SO [X-(3-5i)][X-(3+5i)] SHOULD BE A FACTOR ..THAT IS
(X-3)^2 – (5i)^2 = X^2-6X+9+25 = X^2-6X+34 SHOULD BE A FACTOR. NOW DIVIDE THE POLYNOMIAL WITH THIS
X^2-6X+34] 4X^4 -28X^3 +149X^2-70X-374 [ 4X^2 – 4X –11
4X^4 –24X^3 +136X^2
-4X^3 + 13X^2 –70X
-4X^3 +24X^2 –136X
-11 X^2 +66X -374
-11X^2 +66X –374
0 REMAINDER
HENCE 3-5i IS A FACTOR.
THE REMAINING FACTORS ARE
X = [ 4 + OR – SQRT(4^2+4*4*11)]/(2*4)
= [4+ OR – 8 SQRT (3)]/8
X = (1/2) + OR – SQRT(3)
FACTORS ARE
[X – 0.5 – SQRT(3)] AND [X – 0.5 + SQRT(3)]