Discovering the Term Containing x^{20} in Binomial Theorem (2x - x^4)^{14}

In summary, to find the term containing x^{20} in (2x - x^4)^{14}, we first find the value of k that satisfies 14 + 3k = 20, which is k = 2. Then, we plug in 2 for k and find the third term in the expansion of (2 - x^3)^14, which is (91)(2^12)(-1)^2x^20.
  • #1
ms. confused
91
0
Find the term containing [tex]x^{20}[/tex] in [tex](2x - x^4)^{14}[/tex].

I went [tex]t_{k+1}= _{14}C_{k}(2x)^{14-k}(-x^{4})^{k}[/tex]

[tex]= 2x^{14-k}(-x^{4k})[/tex]

First of all, am I on the right track? If so what exactly do I do from there?
 
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  • #2
Something is definitely wrong. First of all, your binomial coefficient has disappeard on the second line. Secondly, you seem to pull the 2 out of (2x)14-k in an improper way. You also can't bring the k inside the bracket like that, since it's not ((-x)4)k, it's (-x[sup4[/sup])k. Do you see the difference? The exponent 4 applies only to the x, not to (-x), but the exponent k applies to the whole (-x4). You should end up with:

[tex]t_{k+1} = {{14}\choose k}(-2)^{14-k}x^{14 + 3k}[/tex]

See if you can figure out why the above is right. Now, to find the term containing x20, find the value(s) of k that satisfy 14 + 3k = 20. The only value for k is 2. So plug in 2 for k, you'll get:

(91)(212)x20
 
  • #3
Okay I think I get your point about the [tex]-x^4 [/tex] thing. But how did you get the 2x to turn into -2 and how did you get the 14+3k exponent?
 
  • #4
ms. confused said:
Okay I think I get your point about the [tex]-x^4 [/tex] thing. But how did you get the 2x to turn into -2 and how did you get the 14+3k exponent?
You want the x^20 term so if the exponent of x in the kth term is
4k+(14-k)=20 what is k?
 
  • #5
+3k... right...thanks I see it now!
 
  • #6
Take (x^14) common and find the third term in the expansion of (2-(x^3))^14 i.e.
14C2*(2^12)*(-1)^2.
 

What is the Binomial Theorem?

The Binomial Theorem is a mathematical theorem that provides a formula for expanding expressions of the form (a + b)^n, where n is a positive integer. It allows us to easily find the coefficients of each term in the expansion.

What is the relationship between the Binomial Theorem and Pascal's Triangle?

Pascal's Triangle is a geometric arrangement of numbers that can be used to determine the coefficients in the expansion of (a + b)^n. Each row in Pascal's Triangle corresponds to the coefficients of a specific power of (a + b) in the Binomial Theorem.

How do I use the Binomial Theorem to expand a binomial expression?

To use the Binomial Theorem, you first need to determine the values of a, b, and n in the expression (a + b)^n. Then, you can use the formula (a + b)^n = ∑(n choose k)a^(n-k)b^k, where k is the power of b in each term, to find the coefficients of each term in the expansion.

What is the purpose of the Binomial Theorem?

The Binomial Theorem allows us to easily expand binomial expressions without having to manually multiply them out. This is useful in various mathematical and scientific applications, such as in probability, algebra, and calculus.

What are some real-world applications of the Binomial Theorem?

The Binomial Theorem can be used in probability to calculate the chances of certain outcomes in a series of events. It is also used in finance and economics to calculate compound interest and binomial option pricing. In engineering, it can be used to model and predict the behavior of complex systems. Additionally, it has applications in computer science and genetics.

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