Question

Which statement about the simplified binomial expansion of (a + b2)n, where n is a positive integer, is true?

A.The exponent of b will always be even.
B.The exponent of a will always be odd.
C.The sum of the exponents of a and b will always equal n.
D.The sum of the exponents of a and b will always equal n – 1.

Answer 1

We are considering the expansion of the binomial
`( a+b^(2) )^(n)`

since
`( a+b^(2) )^(n) =( a+b^(2) )( a+b^(2) )...( a+b^(2) )` n many times, the first term will be the multiplication of a n times with itself so
`a^(n)`

and the last term will be the multiplication of
`b^(2)` n times with itself that is
`(b^(2)) ^(n)= b^(2n)`

2n, the exponent of b, is even no matter what n is, so

A) is true

B) is not true because if n is odd, the coefficient of a is odd

C) D)

consider the case n=2,


`( a+b^(2) ) ^(2)= a^(2)+2ab^(2)+ b^(4)`

consider the term
`2ab^(2)`, the sum of the exponents of a and be is neither n (2) , nor n-1 (1)



Answer: Only A