Cubic Equation | AMC-10A, 2010 | Problem 21

The polynomial $x^{3}-a x^{2}+b x-2010$ has three positive integer roots. What is the smallest possible value of $a ?$

The given equation is $x^{3}-a x^{2}+b x-2010$

Comparing the equation with \(Ax^3+Bx^2+Cx+D=0\) we get \(A=1,B=-a,C=b,D=0\)

Let us assume that \(x_1,x_2,x_3\) are the roots of the above equation then using vieta's formula we can say that \(x_1.x_2.x_3=2010\)

Therefore if we find out the factors of \(2010\) then we can find out our requirement.....

can you finish the problem........

\(2010\) factors into $2 \cdot 3 \cdot 5 \cdot 67 .$ But, since there are only three roots to the polynomial,two of the four prime factors must be multiplied so that we are left with three roots and we have to find out the smallest positive values of \(a\)

can you finish the problem........

To minimize $a, 2$ and 3 should be multiplied, which means $a$ will be $6+5+67=78$ and the answer is \(78\)

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