Highest common factor

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!

Reply 1

Zacken

22

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?

Reply 2

Substitution

OP

3

Original post by Zacken

Would it not simply be 1?

Maybe, is that because a remainder exists when dividing?

Reply 3

Zacken

22

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. :-)

(edited 8 years ago)

Reply 4

poorform

10

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

Highest common factor

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