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:)
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
Thank you and God bless
To determine the number of factors of 5^x + 2•5^x+1, we first need to simplify the expression.
We can start by factoring out the common factor of 5^x from both terms:
5^x + 2•5^x+1 = 5^x(1 + 2•5) = 5^x(1 + 10)
Simplifying further, we have:
5^x(11)
Now, we have a prime power of 5 (5^x) multiplied by a prime number (11). To find the number of factors, we need to determine the powers of each prime factor in the expression.
For the prime 5, the power is x, which means we have x+1 options for the power of 5 (from 0 to x).
For the prime 11, the power is 1.
To find the total number of factors, we multiply the number of options for each prime factor together.
Therefore, the number of factors of 5^x + 2•5^x+1 is (x+1) * 1, which simplifies to x+1.
So, the answer is B) x + 1.