QUESTIONS - POLYNOMIALS - COMPLEX ROOTS

QUESTIONS-ANSWERS-POLYNOMIALS- COMPLEX ROOTS
On 11/13/06, Anonymous wrote:
> Anonymous has left a new comment on your post "MATHS - 7.INDUCTION":
>
> 1. For a polynomial f(x), suppose that f(12)=0. Then what is one factor of
> f(x)? 2. find a polynomial of degree 4 having zeros 2, i, -2i and

AS PER REMAINDER THEOREM IF F(12)=0 , THEN X-12 IS A FACTOR OF THE FUNCTION F(X).
IF THE POLYNOMIAL HAS REAL COEFFICIENTS THEN ..
IF I IS ZERO,THEN -I IS ALSO A ZERO
SIMILRLY IF -2I IS A ZERO,THEN 2I IS ALSO A ZERO
HENCE THE FACTORS ARE
(X-I),(X+I),(X-2I),(X+2I)....MULTIPLYING THESE FOUR FACTORS WE GET A 4 TH. DEGREE POLYNOMIAL
MULTIPLYING , WE GET
(X^2 - I^2)(X^2 - 4I^2) = (X^2+1)(X^2+4) = X^4+5X^2+4.=P(X)=0
IF X=2 IS ALSO A ZERO THEN WE SHALL NEED A 5 TH. DEGREE POLYNOMIAL...NOT 4 TH. DEGREE POLYNOMIAL.
(X-2)(X^4+5X^2+4) = X^5+5X^3+4X-2X^4-10X^2-8 = X^5-2X^4+5X^3-10X^2+4X-8 =F(X)=0