Binomial Theorem Trick

Added: 2026-05-05

[00:00:00]We are given an expression 2x cubed by 3 minus 1 by 4 x squared whole power 10, and our job is to find the term which is independent of x.

[00:00:12]That means we need to find the term where the power of x becomes zero after expansion.

[00:00:19]Pause this video. Try to solve it on your own, and then let me know your answer in the comments.

[00:00:26]Now, in order to solve this problem, we will use the binomial theorem a + b whole raised to n, whose general formula is given like this.

[00:00:37]First, let us identify the components.

[00:00:40]Here, a is equal to 2x cubed by 3, b is equal to minus 1 by 4 x squared, and n is equal to 10. So, the general term will be combination 10 choose r multiplied by 2x cubed by 3 whole power 10 minus r multiplied by minus 1 by 4 x squared whole power r.

[00:01:05]Now, x squared in the denominator can also be written as x raised to minus 2 whole power r in the numerator.

[00:01:14]Now, focus only on the powers of x. From the first part, we get x raised to 3 times 10 minus r, and from the second part, we get x raised to minus 2r.

[00:01:26]Combining both, the total power of x becomes 30 minus 3r minus 2r, which simplifies to 30 minus 5r.

[00:01:37]For the term to be independent of x, this power must be zero. So, we set 30 minus 5r equal to zero, which gives r equal to 6. The independent term here is t of r + 1, or this means that the independent term is 6 + 1 or seventh term.

[00:01:57]Awesome.

[00:01:58]But hey, you know what? We can also use a nice shortcut or nice trick in order to solve such problems quickly.

[00:02:06]Whenever we have a binomial of the form ax to the power p plus or minus some constant b divided by x to the power q whole raised to power n, and you are asked to find the term independent of x, you can directly use this formula. r equals n multiplied by p divided by p plus q. That's it. For example, in this question, p is 3, q is 2, and n is 10.

[00:02:37]So, directly, r becomes 10 multiplied by 3 divided by 3 plus 2, which is 30 by 5, giving 6. So, without doing the full exponent calculation, we directly found out that the required term is the seventh term.

[00:02:54]This shortcut is extremely useful in exams. Let us see its use in one more example, which is this.

[00:03:01]Root x can also be written as x raised to power half. Here, we have p as 1 by 2, q as 3, and n as 14. Now, directly apply the formula. r equals 14 multiplied by 1 by 2 divided by 1 by 2 plus 3.

[00:03:21]First, simplify the denominator. 1 by 2 plus 3 becomes 7 by 2. Then, the numerator is simply 7. So, now r becomes 7 divided by 7 by 2.

[00:03:34]This simplifies to 2, which means the required term, which is independent of x, is the third term, and that's it.

[00:03:42]It's your turn now. Let me know in the comments which term will be independent of x in this case. Like, share, and subscribe. So good.

[00:03:59]>> [music]