Animated Solution for Mathematics - Basic Mathematics: If $f(x)=x^9-6x^8-2x^7+12x^6+x^4-7x^3+6x^2+x-3$, find $f(6)$.

Imagine you are sitting in the exam hall, and you see this polynomial:

Your heart might skip a beat. It looks like a monster, doesn't it? A degree-nine polynomial!

The immediate, panicked reaction is to start calculating $6^9$, $6^8$, and so on. But stop. Take a deep breath.

In the world of JEE Advanced, whenever you see a high-degree polynomial and a simple integer evaluation, you are not being asked to calculate; you are being asked to observe.

If you try to calculate $6^9$, you are falling right into the trap. The examiners want to see if you have the mathematical maturity to recognize structure.

Let us substitute $x=6$ into our expression:

Now, look at the first two terms: $6^9-6(6^8)$. This is just $6^9-6^9$, which equals $0$. The monster has already started to shrink.

Let us look at the next pair: $-2(6^7)+12(6^6)$. This looks intimidating, but remember that $12=2\cdot 6$.

So, we can rewrite this as:

Since $6\cdot 6^6=6^7$, this becomes $-2(6^7)+2(6^7)$, which is also zero! The pattern is emerging. We are not doing arithmetic; we are performing a dance of cancellations.

Moving to the middle terms: $6^4-7(6^3)+6(6^2)$. We know that $6(6^2)=6^3$.

So, the expression becomes:

Since $6(6^3)=6^4$, we have $6^4-6^4$, which is yet again zero!

After all that, what remains? We have systematically eliminated almost every term.

We are left with the very last part of the polynomial: $x-3$. Substituting $x=6$ into this final, simple expression gives us:

The entire, terrifying degree-nine polynomial collapses into the number 3.

This is the beauty of mathematics—hidden order beneath apparent chaos. Never fear the monster; look for the pattern, and you will find that the solution is often simpler than you ever imagined.