What is the NO-SHORTCUT approach for learning great Mathematics?

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

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

- \(31\)
- \(78\)
- \(43\)

Algebra

Cubic Equation

Roots

But try the problem first...

Answer: \(78\)

Source

Suggested Reading

AMC-10A (2010) Problem 21

Pre College Mathematics

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 \(78\)

- https://www.cheenta.com/problem-based-on-triangle-prmo-2016-problem-10/
- https://www.youtube.com/watch?v=VLyrlx2DWdA&t=17s

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

- What are some of the best colleges for Mathematics that you can aim to apply for after high school?
- How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
- What are the best universities for MS, MMath, and Ph.D. Programs in India?
- What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
- How can you pursue a Ph.D. in Mathematics outside India?
- What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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