Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    1
    ReputationRep:
    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!
    Offline

    22
    ReputationRep:
    (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?
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by Zacken)
    Would it not simply be 1?
    Maybe, is that because a remainder exists when dividing?
    Offline

    22
    ReputationRep:
    (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. :-)
    Offline

    9
    ReputationRep:
    (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
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    What's your favourite Christmas sweets?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.