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When a polynomial f(x) is divided by (x-3) the remainder is -9 and when f(x) is divided by (2x-1) the remainder is -6 . Find the remainder when f(x) is divided by (x-3)(2x-1).

I do not know how to determine the remainder

James Chigwende
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    @MicahWindsor It's elementary abstract algebra often given to illustrate CRT (e.g. see my post in the first link where I show how Lagrange interpolation is a special case of CRT). – Bill Dubuque Apr 29 '20 at 21:53
  • @Gone Thank you for enlightening me! I thought abstract algebra was just rings and fields and that kind of stuff, I didn't realize it got to this level of simplicity. – Micah Windsor Apr 29 '20 at 22:25
  • @MicahWindsor Of course specific concrete instances of abstract theorems generally are simpler, and are often used to motivate the more general abstract results (but, alas, not all textbooks give such motivational examples, e.g. said relationship between CRT and interpolation). – Bill Dubuque Apr 29 '20 at 22:30

3 Answers3

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Hint: $f(x) = A(x) \times (x-3)(2x-1) + Bx + C $.

How can we find $B$ and $C$?

Calvin Lin
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lets say that the reminder is AX+B. So, you have $f(X)=(X-3)(2X-1)+AX+B$. The first statement says that $3A+B=-9$ and the second one says that $1/2A+B=-6 $. Therefore, you have to find $A$ and $B$.

Master.AKA
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Ahmadreza Khazaeipoul
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From

$f(x)=a(x)(x-3)-9\tag-$

$f(x)=b(x)(2x-1)-6\tag-$

we have

$(2x-1)f(x)=a(x)(x-3)(2x-1)-9(2x-1)\tag1$

$(x-3)f(x)=b(x)(2x-1)(x-3)-6(x-3).\tag2$

Multiply $(2)$ by $2$ and subtract from ($1$):

$5f(x)=(a(x)-2b(x))(x-3)(2x-1)-9(2x-1)+12(x-3)$.

Can you take it from here?

J. W. Tanner
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