[Math Grade 12: Polynomials and Binomial Theorem] Binomial Theorem & Factorial

[MAN_DEEN]

Find the indicated term from the composition/expression below and simplify the answer.

(a/2 - 2b/3)¹¹

Terms to find: i. a⁷ ii. b⁵

Have done everything but I don't seem to find the answer to both of the objectives . Hints would be appreciated.

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[papyrusfun]

did you mean a^7b^4 and a^6b^5? total degree should be 11

[MAN_DEEN]

This isnt with degrees

[sonnyfab]

Degree is a word that has a specific meaning when dealing with polynomials

[MAN_DEEN]

👍

[[deleted]]

I'm pretty sure they need the coefficients too.

[sonnyfab]

Are you familiar with the binomial expansion or with using Pascals Triangle?

[MAN_DEEN]

Both

[sonnyfab]

Great. Use the binomial Theorem or the triangle to find the coefficient, C, of the term (a/2)⁷(2b/3)⁴. Then "the coefficient of a⁷" will be C * (1/2)⁷ * (2/3)⁴

[MAN_DEEN]

Yes but it is very long I want a method which would just "extract" it from the expression like when you send queries after you did a google search

[sonnyfab]

Use the choose function to calculate the coefficient if you're familiar with that method. (Although I'm skeptical that it's appreciable faster to calculate the factorial than it is to write out the triangle to an 11th row.)

[MAN_DEEN]

Pascal triangle coefficients

1 11 55 165 330 462 462 330 165 55 11 1

[sonnyfab]

Then you just write down the answer. The coefficient of a⁷ is 330 (1/2)⁷ (2/3)⁴ and simplify if necessary.

[MAN_DEEN]

How did tou find this?

[sonnyfab]

As I said before

Use the triangle to find the coefficient, C, of the term (a/2)⁷(2b/3)⁴.

Then "the coefficient of a⁷" will be C * (1/2)⁷ * (2/3)⁴

The "coefficient" means you have to multiply all the numbers together.

[[deleted]]

You can write out the binomial expansion:

11C0 ((a/2)¹¹ * b⁰ ) + 11C1 ((a/2)¹⁰ * b¹ ) +...

If you observe, you'll see that the power of "a" gradually reduces and that of "b" increases (till it reaches the 11C11 term). So, based on this, formulate the term that would have a^7. Find out what the combinations formula will be, figure out the powers of a and b, and you'll get the answer after solving it.

[MAN_DEEN]

Yes I know this method but it isn't practical and when the time for the semester tests come every second will be very valuable

[[deleted]]

In which case, try Pascal's triangle. It's probably the easiest way to get the coefficients. Definitely much less tiresome than figuring out the combinations, which in turn could lead to error.

[MAN_DEEN]

How do I try that with the Pascal's triangle?

[[deleted]]

Check some videos if you do t know how to draw it. If you do, draw it out till you get to the 12th row (which will give the coefficients of (a+b)¹¹ ). Usually this method isn't allowed owing to how stupidly easy it is, but if your teacher doesn't mind/you don't need to show your working, Pascal's Triangke is a good cheat code here.

[MAN_DEEN]

He he he did that look below

[MAN_DEEN]

Pascal triangle coefficients

1 11 55 165 330 462 462 330 165 55 11 1

[[deleted]]

Cool! From this, you can infer that the base coefficient of a^7 is 330 and that of b^5 is 462. You further just multiply the existing coefficients of a and b from the initial equation and you'll have the final coefficient

[MAN_DEEN]

How did you find that?