Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

When a polynomial $p(x)$ of degree 3 is divided by $3x^2 − 8x + 5$, quotient and remainder obtained are linear polynomials such that $p(1)$ = 19 and $p(5/3)$ = 25. So, find the remainder polynomial.

Please give thorough explanation. I tend to be slow at picking up new things. ;) I'm in 10th grade, so if you use any concept which is above the level of an average 10th grade student, please explain it.

I'll be really grateful.

Thanks :)

share|cite|improve this question
No, both $p(1) = 19$ and $p(5/3) = 25$ are for the dividend polynomial or $p(x)$. – EuclidAteMyBreakfast Jun 30 '14 at 9:36
up vote 0 down vote accepted

There exists a polynomial $q(x)$ such that $$p(x)=(3x^2-8x+5)q(x)+ax+b.$$ Using that $p(1)=19$ you get (note that when you consider $x=1$ you have the equalities $p(1)=(3\cdot 1^2-8\cdot 1+5)\cdot q(1)+a\cdot 1+b=0\cdot q(1)+a+b=a+b$) $$19=a+b$$ and from $p\left(\frac{5}{3}\right)=25$ (using the same argument as before) you have $$25=5a+b.$$

Solve the linear system and you have the solution.

share|cite|improve this answer
@mfl you accidentally took the value of $p(1)$ as 9. It was actually 19, and due to that, Fermat took out the wrong values of "a" and "b". :P But it's okay, I understood what you meant to say. :) The remainder polynomial is $9x + 10$. Thanks a lot. :) – EuclidAteMyBreakfast Jun 30 '14 at 9:30
Opps, I will edit now. @Fermat I have edited to correct a typo. This affects your comment. – mfl Jun 30 '14 at 9:32

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.