AMC 10 Math Olympiad USA Math Olympiad

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

Try this beautiful problem from Algebra, based on the Cubic Equation problem from AMC-10A, 2010. You may use sequential hints to solve the problem.

Try this beautiful problem based on Cubic Equation from AMC 10A, 2010.

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 $a ?$

  • \(31\)
  • \(78\)
  • \(43\)

Key Concepts


Cubic Equation


Check the Answer

But try the problem first…

Answer: \(78\)

Suggested Reading

AMC-10A (2010) Problem 21

Pre College Mathematics

Try with Hints

First hint

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……..

Second Hint

\(2010\) 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……..

Final Step

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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