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19 May 2024, 00:35
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
25%
(medium)
Question Stats:
88% (00:53) correct
11% (01:00) wrong based on 9 sessions
History
Date
Time
Result
Not Attempted Yet
If \(1+x+x^2+x^3=60\), then the average (arithmetic mean) of \(x, x^2, x^3,\) and \(x^4\) is equal to which of the following?
A. 12x
B. 15x
C. 20x
D. 30x
E. 60x
A. 12x
B. 15x
C. 20x
D. 30x
E. 60x
22 Jun 2024, 02:45
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Official Explanation
A quick inspection of the answer choices shows that it is not necessary to solve the equation \(1+x+x^2+x^3=60\) for \(x\) to answer this question. You are being asked to express the average of the four quantities \(x+x^2+x^3\), and \(x^4\) in terms of \(x\). To express this average in terms of \(x\), you need to add the 4 quantities and divide the result by 4; that is, \(\frac{x+x^2+x^3+x^4}{4.}\)
The only information given in the question is that the sum of the 4 quantities, \(1+x+x^2+x^3=60\), is 60, so you need to think of a way to use this information to simplify the expression \(\frac{x+x^2+x^3+x^4}{4.}\)
Note that the numerator of the fraction is a sum of 4 quantities, each of which has an x term raised to a power. Thus, the expression in the numerator can be factored as \(x+x^2+x^3+x^4 = x(1+x+x^2+x^3).\) By using the information in the question, you can make the following simplification.
\(\frac{x+x^2+x^3+x^4}{4}= \frac{x(1+x+x^2+x^3)}{4} = \frac{x(60)}{4} = 15x\)
Therefore, the correct answer is Choice B.
Answer: B
A quick inspection of the answer choices shows that it is not necessary to solve the equation \(1+x+x^2+x^3=60\) for \(x\) to answer this question. You are being asked to express the average of the four quantities \(x+x^2+x^3\), and \(x^4\) in terms of \(x\). To express this average in terms of \(x\), you need to add the 4 quantities and divide the result by 4; that is, \(\frac{x+x^2+x^3+x^4}{4.}\)
The only information given in the question is that the sum of the 4 quantities, \(1+x+x^2+x^3=60\), is 60, so you need to think of a way to use this information to simplify the expression \(\frac{x+x^2+x^3+x^4}{4.}\)
Note that the numerator of the fraction is a sum of 4 quantities, each of which has an x term raised to a power. Thus, the expression in the numerator can be factored as \(x+x^2+x^3+x^4 = x(1+x+x^2+x^3).\) By using the information in the question, you can make the following simplification.
\(\frac{x+x^2+x^3+x^4}{4}= \frac{x(1+x+x^2+x^3)}{4} = \frac{x(60)}{4} = 15x\)
Therefore, the correct answer is Choice B.
Answer: B
