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Natasha2012 [34]
3 years ago
14

Determine the coefficient of x^3 in the expansion of (1 – x)^5(1 + 1/x)^5

Mathematics
1 answer:
Mkey [24]3 years ago
4 0

Notice that

(1 - <em>x</em>)⁵ (1 + 1/<em>x</em>)⁵ = ((1 - <em>x</em>) (1 + 1/<em>x</em>))⁵ = (1 - <em>x</em> + 1/<em>x</em> - 1)⁵ = (1/<em>x</em> - <em>x</em>)⁵

Recall the binomial theorem:

\displaystyle(a+b)^n = \sum_{k=0}^n\binom nk a^{n-k}b^k

Let <em>a</em> = 1/<em>x</em>, <em>b</em> = -<em>x</em>, and <em>n</em> = 5. Then

\displaystyle\left(\frac1x-x\right)^5 = \sum_{k=0}^5\binom5k\left(\frac1x\right)^{5-k}(-x)^k = \sum_{k=0}^5 (-1)^k\binom5k x^{2k-5}

We get an <em>x</em> ³ term for

2<em>k</em> - 5 = 3   ==>   2<em>k</em> = 8   ==>   <em>k</em> = 4

so that the coefficient would be

(-1)^4\dbinom54 = \boxed{5}

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BlackZzzverrR [31]

Answer:

20.922

Step-by-step explanation:

Evaluate ln 7.2 on a calculator; the correct result is 1.974.

Multiply that by 3, obtaining 5.922.

Finally, add 15 to that result:  y = 15 + 3 In 7.2 = 20.922

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3 years ago
The authors of both passages agree that King’s "I Have
Norma-Jean [14]

The authors of both passages agree that King’s "I Have a Dream" speech: had a profound impact on many Americans (Option C).

<h3>What is a comprehension passage?</h3>

In a comprehension passage, some questions are given that follow the passage. The questions are to be answered by using the data given in the passage, even if it differs from real-life facts.

Now,

  • Both the given passages use adjectives like 'magnificent' and 'famous' with respect to Martin Luther King's "I Have a Dream Speech".
  • The first passage claims, "Many people can still remember the first time they heard I Have a Dream, and they tend to speak of that memory with the reverence reserved for a religious experience."
  • While the second concurs with, "For most Americans, those words capture Kings unique genius. They express his immortal longing for freedom, a longing that is familiar to every person who dares imagine a future beyond unjust laws and unfair customs. The edifying universality of those four words who hasn't dreamed, and who cannot identify with people whose dreams of a better world are punished with violence helps to explain their durability."
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To learn more about comprehension passages, refer to the link:

brainly.com/question/24716030

#SPJ4

3 0
1 year ago
In the diagram, point O is the center of the circle and AC¯¯¯¯¯ and BD¯¯¯¯¯ intersect at point E. If m∠AOB = 90° and
Romashka [77]
<span>m∠CED = </span><span>1/2(m∠AOB + </span><span><span>m∠COD)</span> = 1/2(90° + 16°) = 1/2(106°) = 53°</span>

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3 years ago
9+(6w+3w) how do I do this it is to hardThe expression (9+6w)+3w
Mama L [17]

Answer: 9w+9

Step-by-step explanation:

First combine like terms: 6w+3w

You can't combine 9 and 9w so that would be your answer.

Then if it calls for it, use the commutitive property to move 9w before 9.

Hope this helps!

6 0
3 years ago
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LiRa [457]

Tell me if this is fine. I wasn't able to edit the document you gave me, so i made one on my own. If you require a full graph, comment, and ill make it again. (I put 2, select the best one)

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