Question

In the expansion of (3a + 4b)8, which of the following are possible variable terms? Explain your reasoning. a2b3; a5b3; ab8; b8; a4b4; a8; ab7; a6b5

Help please!

Answer 1

We have to find the expansion of
`(3a+4b)^(8)`

We will use binomial expansion to expand the given expression, which states that the expression
`(a+b)^(n)` is expanded as :

`(a+b)^(n)=^(n)C_(0)a^(n)+^(n)C_(1)a^(n-1)b+^(n)C_(2)a^(n-2)b^(2)+........^(n)C_(n)b^(n)`

Now expanding
`(3a+4b)^(8)` we get,

`(3a+4b)^(8)=^(8)C_(0)(3a)^(8)+^(8)C_(1)(3a)^(7)(4b)+^(8)C_(2)(3a)^(6)(4b)^(2)+^(8)C_(3)(3a)^(5)(4b)^(3)+^(8)C_(4)(3a)^(4)(4b)^(4)+^(8)C_(5)(3a)^(3)(4b)^(5)+^(8)C_(6)(3a)^(2)(4b)^(6)+^(8)C_(7)(3a)(4b)^(7)+^(8)C_(8)(4b)^(8)`

`(3a+4b)^(8)=^(8)C_(0)(3)^(8)a^(8)+^(8)C_(1)(3)^(7)(4)(a^(7)b)+^(8)C_(2)(3)^(6)(4)^(2)(a^(6)b^(2))+^(8)C_(3)(3)^(5)(4)^(3)(a^(5)b^(3))+^(8)C_(4)(3)^(4)(4)^(4)(a^(4)b^(4))+^(8)C_(5)(3)^(3)(4)^(5)(a^(3)b^(5))+^(8)C_(6)(3)^(2)(4)^(6)(a^(2)b^(6))+^(8)C_(7)(3)(4)^(7)(ab^(7))+^(8)C_(8)(4)^(8)(b^(8))`

So, the variables are
`a^(5)b^(3)` ,
`b^(8)` ,
`a^(4)b^(4)` ,
`a^(8) , [tex] ab^(7)`