Categories

Roots and coefficients of equations | PRMO 2017 | Question 4

Try this beautiful problem from the Pre-RMO, 2017 based on Roots and coefficients of equations. You may use sequential hints to solve the problem.

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

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.

Subscribe to Cheenta at Youtube

This site uses Akismet to reduce spam. Learn how your comment data is processed.