Pregunta
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The term is a term in which binomial
expansion?

Ask by Cross Morgan. in the United States
Nov 09,2024

Solución de inteligencia artificial de Upstudy

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The term does not appear in either of the given binomial expansions.

Solución

To determine in which binomial expansion the term appears, we will analyze both expressions given.
  1. For the first expression:
    We can start by expanding both the numerator and the denominator.
    • The numerator can be expanded using the binomial theorem:
    • The denominator can also be expanded:
    To find the term , we need to consider the combined effect of both expansions.
    The term in corresponds to :
    where .
    The term in the denominator corresponds to :
    where , , and .
    Thus, the coefficient of in the denominator is:
    The overall coefficient of in the entire expression is:
    Since we are looking for the term , we can see that this term does not appear in this expression.
  2. For the second expression:
    The term cannot appear in since it only contains the terms and .
In conclusion, the term does not appear in either of the given binomial expansions.

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Bonus Knowledge

The term appears in the binomial expansion of . In the expansion, the general term can be expressed as . For , setting (which gives ), we find the coefficient which equals 35, yielding the exact term .
Meanwhile, consider how this ties into the real-world! The binomial expansion isn’t just abstract math—engineers and scientists use it everyday in physics for calculating probabilities, distributions, and even optimizing designs based on varying parameters. So the next time you expand a binomial, remember it’s not just a number crunch; it’s real-world problem-solving at its best!

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