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Roots and coefficients of equations | PRMO 2017 | Question 4

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

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

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

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

$$(-8,-9)$$

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

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