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    ind the quotient and remainder when f(X)=X^3+4⋅X^2+3⋅X+15 is divided by g(X)=X^2+3. Hence, find hcf(f(X),g(X)).

    I calculated the quorient to be x+4 and the remainder 3/(x^2+3) but not sure how to calculate the hcf.

    Thanks!
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    (Original post by Substitution)
    ind the quotient and remainder when f(X)=X^3+4⋅X^2+3⋅X+15 is divided by g(X)=X^2+3. Hence, find hcf(f(X),g(X)).

    I calculated the quorient to be x+4 and the remainder 3/(x^2+3) but not sure how to calculate the hcf.

    Thanks!
    Would it not simply be 1?
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    (Original post by Zacken)
    Would it not simply be 1?
    Maybe, is that because a remainder exists when dividing?
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    (Original post by Substitution)
    Maybe, is that because a remainder exists when dividing?
    I think it's because x^2 + 3 is irreducible and the reason you've given.

    Edit: I'm going to add in a disclaimer that I don't really know what I'm talking about here - so somebody fele free to jump in and correct me. :-)
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    (Original post by Substitution)
    ind the quotient and remainder when f(X)=X^3+4⋅X^2+3⋅X+15 is divided by g(X)=X^2+3. Hence, find hcf(f(X),g(X)).

    I calculated the quorient to be x+4 and the remainder 3/(x^2+3) but not sure how to calculate the hcf.

    Thanks!
    So you have \displaystyle X^3+4X^2+3X+15=(X+4)(X^2+3)+3.

    Now use the following result let \displaystyle f(X),g(X),q(X),r(X) \in \mathbb{F}[X].

    If \displaystyle f(X)=q(X)g(X)+r(X)

    then \displaystyle \text{hcf}(f(X),g(X))=\text{hcf}  (g(X),r(X))

    Now from above we have \displaystyle \text{hcf}(X^3+4X^2+3X+15,X^2+3)  =\text{hcf}(X^2+3,3)=1.

    In fact for \displaystyle f(X) \in \mathbb{F}[X], p \in \mathbb{F}, p \neq 0, ~\text{hcf}(f(X),p)=1
 
 
 
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