Try this beautiful problem from the PRMO, 2017 based on Roots and coefficients of equations.

## Roots and coefficients of equations – PRMO 2017

Let a,b be integers such that all the roots of the equation \((x^{2}+ax+20)(x^{2}+17x+b)\)=0 are negetive integers, find the smallest possible values of a+b.

- is 107
- is 25
- is 840
- cannot be determined from the given information

**Key Concepts**

Polynomials

Roots

Coefficients

## Check the Answer

But try the problem first…

Answer: is 25.

PRMO, 2017, Question 4

Polynomials by Barbeau

## Try with Hints

First hint

\((x^{2}+ax+20)(x^{2}+17x+b)\)

where sum of roots \( \lt \) 0 and product \( \gt 0\) for each quadratic equation \(x^{2}\)+ax+20**=**0 and

\((x^{2}+17x+b)=0\)

\(a \gt 0\), \(b \gt 0\)

now using vieta’s formula on each quadratic equation \(x^{2}\)+ax+20=0 and \((x^{2}+17x+b)=0\), to get possible roots of \(x^{2}\)+ax+20**=0** from product of roots equation \(20=(1 \times 20), (2 \times 10), (4 \times 5)\)

min a=4+5=9 from all sum of roots possible

Second Hint

again using vieta’s formula, to get possible roots of \((x^{2}\)+17x+b)=0 from sum of roots equation \(17=-(\alpha + \beta) \Rightarrow (\alpha,\beta)=(-1,-16),(-2,-15),\)

\((-8,-9)\)

min b=(-1)(-16)=16 from all products of roots possible

Final Step

\((a+b)_{min}=a_{min}+b_{min}\)=9+16=25.

## Other useful links

- https://www.cheenta.com/smallest-perimeter-of-triangle-aime-2015-question-11/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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