Question

A) Write down and simplify the expansion of (a + b/a)^6

, for a, b ∈ R \ 0. (Hint: Use the Binomial Theorem.)

b) What is the coefficient of the b^3 term?

c) Let b = 1. What is the simplified form of the expression now?​

Answer 1

The expansion of
`(a + b/a)^6` using the Binomial Theorem is
`a^6 + 6a^4b + 15a^2b^2 + 20b^3 + 15ab^3/a^2 + 6b^4/a + b^6/a^6.` The coefficient of the
`b^3` term is 20. If b = 1, the expression simplifies to
`a^6 + 6a^4 + 15a^2 + 20 + 15/a^2 + 6/a^4 + 1/a^6.`

Step-by-step explanation:

To expand (a + b/a)^6 using the Binomial Theorem, we need to apply the formula which states that the expansion of (a + b)^n is:

an + nan-1b + n(n-1)an-262/2! + n(n-1)(n-2)an-363/3! + ...

Applying this formula, we get:

`a^6 + 6a^5(b/a) + 15a^4(b/a)^2 + 20a^3(b/a)^3 + 15a^2(b/a)^4 + 6a(b/a)^5 + (b/a)^6`

Simplifying, we have:

`a^6 + 6a^4b + 15a^2b^2 + 20b^3 + 15ab^3/a^2 + 6b^4/a + b^6/a^6`

The coefficient of the
`b^3` term is then 20.

If we let b = 1, the expression simplifies to:

`a^6 + 6a^4 + 15a^2 + 20 + 15/a^2 + 6/a^4 + 1/a^6`

Answer 2

a) a^6+6a^4b+15a^2b^2+20b^3+15(b^4/a^2)+6(b^5/a^4)+(b/a)^6

b) 20

c) a^6+6a^4+15a^2+20+15/a^2+6/a^4+1/a^6

Step-by-step explanation:

(a+b/a)^6=a^6+6a^5(b/a)+15a^4(b/a)^2+20a^3(b/a)^3+15a^2(b/a)^4+6a(b/a)^5+(b/a)^6

a^6+6a^4b+15a^2b^2+20b^3+15(b^4/a^2)+6(b^5/a^4)+(b/a)^6

b) the coefficient of b^3=20

c) if b=1, the expression is

a^6+6a^4+15a^2+20+15/a^2+6/a^4+1/a^6