What Is The Fifth Term In The Binomial Expansion Of $(x+5)^8$?A. $175,000 X^3$B. \$43,750 X^4$[/tex\]C. $3,125 X^5$D. $7,000 X^5$

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Introduction

The binomial expansion is a mathematical concept that allows us to expand expressions of the form $(a+b)^n$, where aa and bb are numbers or variables, and nn is a positive integer. The expansion is a sum of terms, each of which is a combination of powers of aa and bb. In this article, we will explore the binomial expansion of the expression $(x+5)^8$ and find the fifth term.

The Binomial Theorem

The binomial theorem is a formula that describes the expansion of $(a+b)^n$. It is given by:

(a+b)n=(n0)anb0+(n1)an−1b1+(n2)an−2b2+⋯+(nn−1)a1bn−1+(nn)a0bn(a+b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n

where (nk)\binom{n}{k} is the binomial coefficient, defined as:

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

Expanding $(x+5)^8$

To find the fifth term in the expansion of $(x+5)^8$, we need to apply the binomial theorem. We will use the formula above to expand the expression.

The first few terms of the expansion are:

(x+5)8=(80)x8(5)0+(81)x7(5)1+(82)x6(5)2+(83)x5(5)3+(84)x4(5)4+⋯(x+5)^8 = \binom{8}{0} x^8 (5)^0 + \binom{8}{1} x^7 (5)^1 + \binom{8}{2} x^6 (5)^2 + \binom{8}{3} x^5 (5)^3 + \binom{8}{4} x^4 (5)^4 + \cdots

Finding the Fifth Term

To find the fifth term, we need to find the coefficient of x5x^5 in the expansion. We can do this by looking at the term with x5x^5:

(83)x5(5)3=56x5(125)\binom{8}{3} x^5 (5)^3 = 56 x^5 (125)

Simplifying the Term

To simplify the term, we can multiply the coefficient by the power of 55:

56x5(125)=7,000x556 x^5 (125) = 7,000 x^5

Conclusion

In conclusion, the fifth term in the binomial expansion of $(x+5)^8$ is $7,000 x^5$. This is the correct answer among the options provided.

Discussion

The binomial expansion is a powerful tool for expanding expressions of the form $(a+b)^n$. It is used in many areas of mathematics, including algebra, geometry, and calculus. The binomial theorem is a fundamental concept in mathematics, and it has many applications in science and engineering.

Final Answer

The final answer is: 7,000x5\boxed{7,000 x^5}

Introduction

In our previous article, we explored the binomial expansion of the expression $(x+5)^8$ and found the fifth term. In this article, we will answer some common questions related to binomial expansion and the fifth term.

Q: What is the binomial expansion?

A: The binomial expansion is a mathematical concept that allows us to expand expressions of the form $(a+b)^n$, where aa and bb are numbers or variables, and nn is a positive integer. The expansion is a sum of terms, each of which is a combination of powers of aa and bb.

Q: How do I apply the binomial theorem?

A: To apply the binomial theorem, you need to use the formula:

(a+b)n=(n0)anb0+(n1)an−1b1+(n2)an−2b2+⋯+(nn−1)a1bn−1+(nn)a0bn(a+b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n

where (nk)\binom{n}{k} is the binomial coefficient, defined as:

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

Q: What is the binomial coefficient?

A: The binomial coefficient is a number that appears in the binomial expansion. It is defined as:

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

Q: How do I find the fifth term in the binomial expansion?

A: To find the fifth term, you need to look at the term with x5x^5 in the expansion. You can do this by using the binomial theorem and finding the coefficient of x5x^5.

Q: What is the fifth term in the binomial expansion of $(x+5)^8$?

A: The fifth term in the binomial expansion of $(x+5)^8$ is $7,000 x^5$.

Q: What is the binomial theorem used for?

A: The binomial theorem is used in many areas of mathematics, including algebra, geometry, and calculus. It is also used in science and engineering to solve problems involving binomial expansions.

Q: Can I use the binomial theorem to expand expressions with negative exponents?

A: Yes, you can use the binomial theorem to expand expressions with negative exponents. However, you need to be careful when dealing with negative exponents, as they can lead to complex expressions.

Q: How do I simplify the binomial expansion?

A: To simplify the binomial expansion, you can use the binomial theorem and combine like terms. You can also use algebraic manipulations to simplify the expression.

Q: What are some common applications of the binomial theorem?

A: Some common applications of the binomial theorem include:

  • Expanding expressions of the form $(a+b)^n$
  • Finding the sum of a series
  • Solving problems involving binomial expansions
  • Calculating probabilities and statistics

Conclusion

In conclusion, the binomial expansion is a powerful tool for expanding expressions of the form $(a+b)^n$. The binomial theorem is a fundamental concept in mathematics, and it has many applications in science and engineering. We hope that this Q&A article has helped you understand the binomial expansion and the fifth term.

Final Answer

The final answer is: 7,000x5\boxed{7,000 x^5}