Type The Correct Answer In The Box. Use Numerals Instead Of Words.Fill In The Missing Coefficients Of The Terms In The Expansion Of { (x+y)^9$} . . . [ X^9 + 9x^8y + \square X 7y 2 + 84x 6y 3 + \square X 5y 4 + 126x 4y 5 + \square X 3y 6 +

The Binomial Theorem: Expanding ${(x+y)^9}$

The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form ${(a+b)^n}$, where ${a}$ and ${b}$ are any real numbers and ${n}$ is a positive integer. In this article, we will use the binomial theorem to expand the expression ${(x+y)^9}$ and fill in the missing coefficients of the terms.

Understanding the Binomial Theorem

The binomial theorem states that for any positive integer ${n}$, the expansion of ${(a+b)^n}$ is given by:

${(a+b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n-1}ab^{n-1} + \binom{n}{n}b^n}$

where ${\binom{n}{k}}$ is the binomial coefficient, defined as:

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

Expanding ${(x+y)^9}$

To expand ${(x+y)^9}$, we can use the binomial theorem with ${a=x}$, ${b=y}$, and ${n=9}$. The expansion will have 10 terms, each with a binomial coefficient and a power of ${x}$ and ${y}$.

The first few terms of the expansion are:

${(x+y)^9 = \binom{9}{0}x^9 + \binom{9}{1}x^8y + \binom{9}{2}x^7y^2 + \cdots}$

We are given the first four terms of the expansion, and we need to fill in the missing coefficients.

Filling in the Missing Coefficients

To fill in the missing coefficients, we can use the binomial theorem formula and calculate the binomial coefficients for each term.

The first term is ${\binom{9}{0}x^9 = x^9}$, which has a coefficient of 1.

The second term is ${\binom{9}{1}x^8y = 9x^8y}$, which has a coefficient of 9.

The third term is ${\binom{9}{2}x^7y^2 = \square x^7y^2}$, which has a missing coefficient.

The fourth term is ${\binom{9}{3}x^6y^3 = 84x^6y^3}$, which has a coefficient of 84.

The fifth term is ${\binom{9}{4}x^5y^4 = \square x^5y^4}$, which has a missing coefficient.

The sixth term is ${\binom{9}{5}x^4y^5 = 126x^4y^5}$, which has a coefficient of 126.

The seventh term is ${\binom{9}{6}x^3y^6 = \square x^3y^6}$, which has a missing coefficient.

The eighth term is ${\binom{9}{7}x^2y^7 = 126x^2y^7}$, which has a coefficient of 126.

The ninth term is ${\binom{9}{8}xy^8 = 63xy^8}$, which has a coefficient of 63.

The tenth term is ${\binom{9}{9}y^9 = 1y^9}$, which has a coefficient of 1.

Calculating the Missing Coefficients

To calculate the missing coefficients, we can use the binomial theorem formula and calculate the binomial coefficients for each term.

For the third term, ${\binom{9}{2}x^7y^2 = \square x^7y^2}$, we have:

${\binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9!}{2!7!} = \frac{9 \times 8}{2} = 36}$

So, the third term is ${36x^7y^2}$.

For the fifth term, ${\binom{9}{4}x^5y^4 = \square x^5y^4}$, we have:

${\binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9!}{4!5!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2} = 126}$

So, the fifth term is ${126x^5y^4}$.

For the seventh term, ${\binom{9}{6}x^3y^6 = \square x^3y^6}$, we have:

${\binom{9}{6} = \frac{9!}{6!(9-6)!} = \frac{9!}{6!3!} = \frac{9 \times 8 \times 7}{3 \times 2} = 84}$

So, the seventh term is ${84x^3y^6}$.

Conclusion

In this article, we used the binomial theorem to expand the expression ${(x+y)^9}$ and fill in the missing coefficients of the terms. We calculated the binomial coefficients for each term and found that the missing coefficients are 36, 126, and 84.

The complete expansion of ${(x+y)^9}$ is:

${(x+y)^9 = x^9 + 9x^8y + 36x^7y^2 + 84x^6y^3 + 126x^5y^4 + 126x^4y^5 + 84x^3y^6 + 126x^2y^7 + 63xy^8 + y^9}$

We hope this article has helped you understand the binomial theorem and how to use it to expand expressions of the form ${(a+b)^n}$.
Q&A: The Binomial Theorem and Expanding ${(x+y)^9}$

In our previous article, we used the binomial theorem to expand the expression ${(x+y)^9}$ and fill in the missing coefficients of the terms. In this article, we will answer some common questions about the binomial theorem and expanding expressions of the form ${(a+b)^n}$.

Q: What is the binomial theorem?

A: The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form ${(a+b)^n}$, where ${a}$ and ${b}$ are any real numbers and ${n}$ is a positive integer.

Q: How do I use the binomial theorem to expand an expression?

A: To use the binomial theorem to expand an expression, you need to follow these steps:

1. Identify the values of ${a}$, ${b}$, and ${n}$ in the expression.
2. Use the binomial theorem formula to calculate the binomial coefficients for each term.
3. Write out the terms of the expansion, using the binomial coefficients and the powers of ${a}$ and ${b}$.

Q: What is a binomial coefficient?

A: A binomial coefficient is a number that appears in the binomial theorem formula. It is calculated as:

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

where ${n}$ is the exponent and ${k}$ is the term number.

Q: How do I calculate a binomial coefficient?

A: To calculate a binomial coefficient, you can use the formula above. For example, to calculate ${\binom{9}{2}}$, you would use:

${\binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9!}{2!7!} = \frac{9 \times 8}{2} = 36}$

Q: What is the difference between a binomial coefficient and a factorial?

A: A binomial coefficient is a number that appears in the binomial theorem formula, while a factorial is the product of all positive integers up to a given number. For example, ${5!}$ is the product of all positive integers up to 5:

${5! = 5 \times 4 \times 3 \times 2 \times 1 = 120}$

Q: How do I use the binomial theorem to expand a negative exponent?

A: To use the binomial theorem to expand a negative exponent, you need to follow these steps:

1. Rewrite the expression with a positive exponent.
2. Use the binomial theorem formula to calculate the binomial coefficients for each term.
3. Write out the terms of the expansion, using the binomial coefficients and the powers of ${a}$ and ${b}$.

Q: What is the relationship between the binomial theorem and the Pascal's triangle?

A: The binomial theorem is closely related to Pascal's triangle, which is a triangular array of numbers that can be used to calculate binomial coefficients. Each number in Pascal's triangle is the sum of the two numbers directly above it.

Q: How do I use the binomial theorem to solve problems in algebra?

A: The binomial theorem can be used to solve a wide range of problems in algebra, including:

• Expanding expressions of the form ${(a+b)^n}$
• Calculating binomial coefficients
• Solving equations and inequalities involving binomial expressions
• Finding the roots of polynomials involving binomial expressions

Conclusion

In this article, we have answered some common questions about the binomial theorem and expanding expressions of the form ${(a+b)^n}$. We hope this article has helped you understand the binomial theorem and how to use it to solve problems in algebra.