Question

Determine the number of factors of 5^x + 2•5^x+1.

Choices:
A) x
B) x + 1
C) 2x
D) 2x + 2

Please show solution:)

Answer 1

The way you typed it : 5^x + 2•5^x+1
= 3*5^x + 1

it does not factor over the rational numbers
Secondly you did not say what type of number x is, let's assume x is a whole number, that is x = 0, 1, 2, 3, ...
if you meant:
5^x + 2*5^(x+1)

= 5^x( 1 + 2(5^1) )
= 5^x ( 11)
or
11*5^x

So now it depends on the value of x
e.g. if x = 3, we have, 5^3 + 2*5^4 = 1345, which is the same as my
11*5*5*5 , or 1345

the 11 can be picked in 2 ways, either we use it or we don't
the 5 can be used in 4 ways, we can use it once twice or three times or none at all
so the number of factors of 1345 is 2(4) or 8 ways,
BUT, that includes not taking the 11 or any 5,
So if x = 3 we have 8-1 or 7 factors.

if we have 11*5^x, the number of factors would be
(2)(x+1) - 1
= 2x + 2 - 1
= 2x + 1

I don't see that choice, and I don't see any flaw in my analysis. Since 1 is a factor for every number it is usually not included in the list of factors.
If we include that exception, then it would 2x

Check:
factors of 1345:
5, 11, 25, 55, 125, 275, 1345 ----> 7 of them

Answer 2

Thank you and God bless