220k views
21 votes
Write down the first three and last two terms of the binomial expansion of:

a) (3x+(2/x))^15
b) (2x-(3/x))^20

1 Answer

6 votes

Answer:

a)
(3x)^(15) + 15(3x)^(14)((2)/(x)) + 105(3x)^(13)((2)/(x))^2 + ... + 15(3x)((2)/(x))^(14) + ((2)/(x))^(15)

b)
(2x)^(20) + 20(2x)^(19)(-(3)/(x)) + 190(2x)^(18)(-(3)/(x))^2 + ... + 20(2x)(-(3)/(x))^(19) + (-(3)/(x))^(20)

Explanation:

The binomial expansion formula is:

(a + b)^n = a^n + \binom{n}{1}a^(n-1) b + \binom{n}{2}a^(n-2)b^2 + ... + \binom{n}{r}a^(n-r)b^(r) + ... + b^n
The (n r) in front of each term is the binomial coefficient. This can be calculated on a calculator using the nCr button (in this case, you'd put 15C1 for (n 1), 15C2 for (n 2), etc). This can be calculated without a calculator using this formula:

\binom{n}{r} = \frac{{n!}}{{r!\left( {n - r} \right)!}}

In a), a = 3x, b = 2/x and n = 15.

You can plug these into the formula above to get:

(3x)^(15) + 15(3x)^(14)((2)/(x)) + 105(3x)^(13)((2)/(x))^2 + ... + 15(3x)((2)/(x))^(14) + ((2)/(x))^(15)

Here's how to do this without the formula:

  1. Start with the a^n
  2. In the second term, you subtract 1 from a's power (in this case, 15 to 14) and add 1 to b's power (0 to 1). Then you multiply it by nCr.
  3. This keeps going until you get to b^n. To check you've done it correctly, the powers of a and b in each term should add up to n. For example, in term 3 above, a's power is 13 and b's power is 2. These add to make 15.
  4. Remember, a's power goes down by 1 each time, and b's power goes up by 1 each time. There is no b in the first term because b^0 = 1.

You could 'simplify' this by expanding the brackets, but this gives you massive numbers so it's probably best to leave it like this.

Applying this to b):

a = 2x, b = -3/x, n = 20



(2x)^(20) + 20(2x)^(19)(-(3)/(x)) + 190(2x)^(18)(-(3)/(x))^2 + ... + 20(2x)(-(3)/(x))^(19) + (-(3)/(x))^(20)

User Grumpy Cat
by
8.9k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories