Find the remainder without differentiation

Asked Aug 15 '23 at 02:53

Active Aug 15 '23 at 03:43

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The degree of $f(x)$ is not given. When $f(x)$ is divided by $(x-3)$ the remainder is $15$ and when $f(x)$ is divided by $(x-1)^2$ the remainder is $2x+1$.

Now we need to find the remainder when $f(x)$ is divided by $(x-1)^2(x-3)$.

My attempt : I used remainder theorem and differentiation to obtain the answer $2x^2-2x+3$

Can we find the remainder without using differentiation?

edited Aug 15 '23 at 03:43

Ted Shifrin

115,160

asked Aug 15 '23 at 02:53

Angelo Mark

5,954

1

Have you tried the Chinese Remainder Theorem? (I added that tag.) – Ted Shifrin Aug 15 '23 at 03:42
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Does this answer your question? Finding the remainder when a polynomial is divided by a product of numbers whose remainders are known - found using an Approach0 search. Note there are also quite a few other duplicates found, e.g., Remainders and polynomial division ... – John Omielan Aug 15 '23 at 04:10
(cont) (with the answer there including a helpful link to Wikipedia's Polynomial remainder theorem article), Remainder of the polynomial, Finding remainder, Remainder for the polynomial division (although the answer there using differentiation), Polynomial remainder problem. (with its answer also using differentiation), etc. Note there are also several AoPS threads, e.g., ... – John Omielan Aug 15 '23 at 04:17
(cont.) Algebra......., Crazy Polynomials (where the first of the current $2$ solutions uses differentiation), etc. – John Omielan Aug 15 '23 at 04:19

Find the remainder without differentiation

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