M31-04

Question

If  1 + x^4 + x^3 + x^2 + x = 80 , what is the average (arithmetic mean) of  x ,  x^2 ,  x^3 ,  x^4 , and  x^5 ?

A.  12x 
B.  13x 
C.  14x 
D.  16x 
E.  20x

Bunuel · Answer

Official Solution:

If  1 + x^4 + x^3 + x^2 + x = 80 , what is the average (arithmetic mean) of  x ,  x^2 ,  x^3 ,  x^4 , and  x^5 ?

A.  12x 
B.  13x 
C.  14x 
D.  16x 
E.  20x

The average (arithmetic mean) of  x ,  x^2 ,  x^3 ,  x^4 , and  x^5  is given by  \frac{x + x^2 + x^3 + x^4 + x^5}{5} . Factoring out  x , we get  \frac{x(1 + x + x^2 + x^3 + x^4)}{5} = \frac{80x}{5} = 16x .

Answer: D

marcoelmanchi · Answer

Hey Bunuel!

I do not see how x(1+x+x^2+x^3+x^4) = 80x. Can you please explain?

Many thanks!

Bunuel · Answer

We are given that  1 + x^4 + x^3 + x^2 + x = 80 , thus  x(1+x+x^2+x^3+x^4) = x*80=80x .

theeni03 · Answer

I think this is a  high-quality  question and I agree with explanation.

Rebaz · Answer

This is how i approached this question:

Let say , If X=10, then what is the value of 5X? So you are given the value of X , and then you are asked about the value of 5X. These are two separate pieces of information and this is important to know.

Thus, If X=10 then 5X=5(10)= 50 is the the value of 5X.

Let's discuss the original question.

If 1+ x^4 +x^3 + x^2 + x = 80, then what is the average of x, x^2 +x^3 + x^4, x^5? so we have here two separate pieces of information as well.

We know that the formula for Average: Average=Sum of the terms/number of the terms.

Thus, Average= x, x^2 +x^3 + x^4, x^5 / 5(number of the terms) = but here is the trick, we can simplify the sum of the terms by pulling out the common factors, and if we do that, then we see that x, x^2 +x^3 + x^4, x^5 be comes X(1+x+x^2+x^3+x^4). Since from that question stem (1+x+x^2+x^3+x^4)= 80 , then simply substitute 80 for (1+x+x^2+x^3+x^4), then the avergae becomes X(80) / 5(number of the terms)= 16X

The correct answer is D

I hope it's clear.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

sssdd1234 · Answer

I think this is a  high-quality  question.

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M31-04

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