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 :)

  • $\begingroup$ No, both $p(1) = 19$ and $p(5/3) = 25$ are for the dividend polynomial or $p(x)$. $\endgroup$ – EuclidAteMyBreakfast Jun 30 '14 at 9:36

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.

  • $\begingroup$ @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. :) $\endgroup$ – EuclidAteMyBreakfast Jun 30 '14 at 9:30
  • $\begingroup$ Opps, I will edit now. @Fermat I have edited to correct a typo. This affects your comment. $\endgroup$ – mfl Jun 30 '14 at 9:32

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