MTH120 College Algebra Chapter 9.6

 9.6 Binomial Theorem

The Binomial Theorem is a fundamental result in algebra that provides an expression for expanding the powers of a binomial, which is an algebraic expression with two terms. The Binomial Theorem is particularly useful for finding the expansion of expressions in the form $(�+�{)}^{�}$, where $�$ and $�$ are constants and $�$ is a non-negative integer.

The Binomial Theorem states that for any non-negative integer $�$, the expansion of $(�+�{)}^{�}$ can be expressed as a sum of terms, each of which is a multiple of ${�}^{�}$ and ${�}^{�}$, where $�$ and $�$ are non-negative integers that satisfy the condition $�+�=�$. This expansion is given by:

$(�+�{)}^{�}=\left(\genfrac{}{}{0px}{}{�}{0}\right){�}^{�}{�}^0+\left(\genfrac{}{}{0px}{}{�}{1}\right){�}^{�-1}{�}^1+\left(\genfrac{}{}{0px}{}{�}{2}\right){�}^{�-2}{�}^2+\dots +\left(\genfrac{}{}{0px}{}{�}{�}\right){�}^0{�}^{�}$

Where:

• $\left(\genfrac{}{}{0px}{}{�}{�}\right)$ represents the binomial coefficient, which is equal to $\frac{�!}{�!(�-�)!}$.
• $�!$ is the factorial of $�$, which is the product of all positive integers from 1 to $�$.

In this expansion, the first term has ${�}^{�}$ and no $�$ terms (${�}^0$), and the last term has no $�$ terms (${�}^0$) and ${�}^{�}$. The remaining terms have combinations of $�$ and $�$ raised to various powers.

The Binomial Theorem is a powerful tool for expanding expressions, simplifying calculations, and finding coefficients in polynomial expressions. It is widely used in algebra, combinatorics, and calculus.

Binomial coefficients, denoted as $\left(\genfrac{}{}{0px}{}{�}{�}\right)$, represent the number of ways to choose $�$ items from a set of $�$ items without regard to the order. Binomial coefficients are also known as "combinations." They are commonly used in the Binomial Theorem and in combinatorics. Here are some examples of how to identify binomial coefficients:

Example 1: Find the binomial coefficient $\left(\genfrac{}{}{0px}{}{5}{2}\right)$.

This binomial coefficient represents the number of ways to choose 2 items from a set of 5 items without regard to order. It can be calculated using the formula:

$\left(\genfrac{}{}{0px}{}{5}{2}\right)=\frac{5!}{2!(5-2)!}=\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{2\cdot 1\cdot 3\cdot 2\cdot 1}=10$

So, $\left(\genfrac{}{}{0px}{}{5}{2}\right)=10$, which means there are 10 ways to choose 2 items from a set of 5.

Example 2: Find the binomial coefficient $\left(\genfrac{}{}{0px}{}{7}{4}\right)$.

This binomial coefficient represents the number of ways to choose 4 items from a set of 7 items without regard to order. It can be calculated using the formula:

$\left(\genfrac{}{}{0px}{}{7}{4}\right)=\frac{7!}{4!(7-4)!}=\frac{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{4\cdot 3\cdot 2\cdot 1\cdot 3\cdot 2\cdot 1}=35$

So, $\left(\genfrac{}{}{0px}{}{7}{4}\right)=35$, which means there are 35 ways to choose 4 items from a set of 7.

Example 3: Find the binomial coefficient $\left(\genfrac{}{}{0px}{}{10}{0}\right)$.

This binomial coefficient represents the number of ways to choose 0 items from a set of 10 items without regard to order. In this case, it's important to note that there is only one way to choose 0 items, which is by not choosing anything. Therefore:

$\left(\genfrac{}{}{0px}{}{10}{0}\right)=1$

Binomial coefficients can be calculated using the formula $\frac{�!}{�!(�-�)!}$, and they represent the number of combinations or ways to select $�$ items from a set of $�$ items. In some cases, the answer may be 1, as shown in Example 3, which signifies that there's only one way to make that selection.

The Binomial Theorem is a powerful tool for expanding binomial expressions, and it allows you to find the expansion of expressions in the form $(�+�{)}^{�}$. The expansion is represented as a sum of terms, and each term consists of a binomial coefficient, one of the variables (usually $�$), and the other variable (usually $�$) raised to a certain power. Here are some examples of how to use the Binomial Theorem:

Example 1: Expand $(�+�{)}^3$.

The Binomial Theorem expansion for this expression is:

$(�+�{)}^3=\left(\genfrac{}{}{0px}{}{3}{0}\right){�}^3{�}^0+\left(\genfrac{}{}{0px}{}{3}{1}\right){�}^2{�}^1+\left(\genfrac{}{}{0px}{}{3}{2}\right){�}^1{�}^2+\left(\genfrac{}{}{0px}{}{3}{3}\right){�}^0{�}^3$

Now, calculate the binomial coefficients:

• $\left(\genfrac{}{}{0px}{}{3}{0}\right)=1$
• $\left(\genfrac{}{}{0px}{}{3}{1}\right)=3$
• $\left(\genfrac{}{}{0px}{}{3}{2}\right)=3$
• $\left(\genfrac{}{}{0px}{}{3}{3}\right)=1$

So, the expansion is:

$(�+�{)}^3={�}^3+3{�}^2�+3�{�}^2+{�}^3$

Example 2: Expand $(�-�{)}^4$.

The Binomial Theorem expansion for this expression is:

$(�-�{)}^4=\left(\genfrac{}{}{0px}{}{4}{0}\right){�}^4(-�{)}^0+\left(\genfrac{}{}{0px}{}{4}{1}\right){�}^3(-�{)}^1+\left(\genfrac{}{}{0px}{}{4}{2}\right){�}^2(-�{)}^2+\left(\genfrac{}{}{0px}{}{4}{3}\right){�}^1(-�{)}^3+\left(\genfrac{}{}{0px}{}{4}{4}\right){�}^0(-�{)}^4$

Now, calculate the binomial coefficients:

• $\left(\genfrac{}{}{0px}{}{4}{0}\right)=1$
• $\left(\genfrac{}{}{0px}{}{4}{1}\right)=4$
• $\left(\genfrac{}{}{0px}{}{4}{2}\right)=6$
• $\left(\genfrac{}{}{0px}{}{4}{3}\right)=4$
• $\left(\genfrac{}{}{0px}{}{4}{4}\right)=1$

So, the expansion is:

$(�-�{)}^4={�}^4-4{�}^3�+6{�}^2{�}^2-4�{�}^3+{�}^4$

Example 3: Expand $(2�-3�{)}^5$.

The Binomial Theorem expansion for this expression is:

$(2�-3�{)}^5=\left(\genfrac{}{}{0px}{}{5}{0}\right)(2�{)}^5(-3�{)}^0+\left(\genfrac{}{}{0px}{}{5}{1}\right)(2�{)}^4(-3�{)}^1+\left(\genfrac{}{}{0px}{}{5}{2}\right)(2�{)}^3(-3�{)}^2+\left(\genfrac{}{}{0px}{}{5}{3}\right)(2�{)}^2(-3�{)}^3+\left(\genfrac{}{}{0px}{}{5}{4}\right)(2�{)}^1(-3�{)}^4+\left(\genfrac{}{}{0px}{}{5}{5}\right)(2�{)}^0(-3�{)}^5$

Now, calculate the binomial coefficients:

• $\left(\genfrac{}{}{0px}{}{5}{0}\right)=1$
• $\left(\genfrac{}{}{0px}{}{5}{1}\right)=5$
• $\left(\genfrac{}{}{0px}{}{5}{2}\right)=10$
• $\left(\genfrac{}{}{0px}{}{5}{3}\right)=10$
• $\left(\genfrac{}{}{0px}{}{5}{4}\right)=5$
• $\left(\genfrac{}{}{0px}{}{5}{5}\right)=1$

So, the expansion is:

$(2�-3�{)}^5=32{�}^5-240{�}^4�+720{�}^3{�}^2-1080{�}^2{�}^3+810�{�}^4-243{�}^5$

The Binomial Theorem is a useful tool for simplifying and expanding expressions involving binomials. It can be applied to various situations where you need to find the expansion of such expressions.

To use the Binomial Theorem to find a single term in the expansion of $(�+�{)}^{�}$, you can apply the formula for a specific term. The term in the expansion is determined by a binomial coefficient $\left(\genfrac{}{}{0px}{}{�}{�}\right)$, where $�$ is the term's position, and the powers of $�$ and $�$ associated with that term. Here's how to find a single term using the Binomial Theorem:

The formula for the $�$-th term in the expansion of $(�+�{)}^{�}$ is:

${�}_{�}=\left(\genfrac{}{}{0px}{}{�}{�}\right){�}^{�-�}{�}^{�}$

Where:

• ${�}_{�}$ is the $�$-th term in the expansion.
• $\left(\genfrac{}{}{0px}{}{�}{�}\right)$ is the binomial coefficient, which can be calculated as $\frac{�!}{�!(�-�)!}$.
• ${�}^{�-�}$ represents $�$ raised to the power of $�-�$.
• ${�}^{�}$ represents $�$ raised to the power of $�$.

Here are some examples of finding single terms in the expansion:

Example 1: Find the 3rd term in the expansion of $(�+2�{)}^5$.

In this case, $�=5$ (the power), and you want to find the 3rd term ($�=3$) in the expansion. Apply the formula:

${�}_3=\left(\genfrac{}{}{0px}{}{5}{3}\right){�}^{5-3}(2�{)}^3$

Calculate the binomial coefficient:

$\left(\genfrac{}{}{0px}{}{5}{3}\right)=\frac{5!}{3!(5-3)!}=\frac{5!}{3!\cdot 2!}=\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{3\cdot 2\cdot 1\cdot 2\cdot 1}=10$

Now, calculate the powers of $�$ and $2�$:

${�}^{5-3}={�}^2$ $(2�{)}^3=2^3{�}^3=8{�}^3$

Combine the results:

${�}_3=10{�}^2(8{�}^3)=80{�}^2{�}^3$

So, the 3rd term in the expansion of $(�+2�{)}^5$ is $80{�}^2{�}^3$.

Example 2: Find the 4th term in the expansion of $(�-�{)}^6$.

In this case, $�=6$, and you want to find the 4th term ($�=4$) in the expansion. Apply the formula:

${�}_4=\left(\genfrac{}{}{0px}{}{6}{4}\right){�}^{6-4}(-�{)}^4$

Calculate the binomial coefficient:

$\left(\genfrac{}{}{0px}{}{6}{4}\right)=\frac{6!}{4!(6-4)!}=\frac{6!}{4!\cdot 2!}=\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{4\cdot 3\cdot 2\cdot 1\cdot 2\cdot 1}=15$

Now, calculate the powers of $�$ and $-�$:

${�}^{6-4}={�}^2$ $(-�{)}^4=(-1{)}^4{�}^4={�}^4$

Combine the results:

${�}_4=15{�}^2{�}^4$

So, the 4th term in the expansion of $(�-�{)}^6$ is $15{�}^2{�}^4$.

In each of these examples, the Binomial Theorem formula for a specific term was used to find the desired term in the expansion.

To find the $(�+1)$-th term of a binomial expansion, such as $(�+�{)}^{�}$, you can use the Binomial Theorem. The $(�+1)$-th term is determined by the binomial coefficient $\left(\genfrac{}{}{0px}{}{�}{�}\right)$ and the powers of $�$ and $�$ associated with that term.

The formula for the $(�+1)$-th term in the expansion of $(�+�{)}^{�}$ is:

${�}_{�+1}=\left(\genfrac{}{}{0px}{}{�}{�}\right){�}^{�-�}{�}^{�}$

Where:

• ${�}_{�+1}$ is the $(�+1)$-th term in the expansion.
• $\left(\genfrac{}{}{0px}{}{�}{�}\right)$ is the binomial coefficient, which can be calculated as $\frac{�!}{�!(�-�)!}$.
• ${�}^{�-�}$ represents $�$ raised to the power of $�-�$.
• ${�}^{�}$ represents $�$ raised to the power of $�$.

To find the $(�+1)$-th term, you need to know the values of $�$ (the power), $�$ (the term you want to find), and the coefficients of $�$ and $�$ in the expansion.

Here's an example:

Example: Find the 5th term in the expansion of $(�-3�{)}^7$.

In this case, $�=7$ (the power), and you want to find the 5th term ($�=4$) in the expansion. Apply the formula:

${�}_5=\left(\genfrac{}{}{0px}{}{7}{4}\right){�}^{7-4}(-3�{)}^4$

Calculate the binomial coefficient:

$\left(\genfrac{}{}{0px}{}{7}{4}\right)=\frac{7!}{4!(7-4)!}=\frac{7!}{4!\cdot 3!}=\frac{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{4\cdot 3\cdot 2\cdot 1\cdot 3\cdot 2\cdot 1}=35$

Now, calculate the powers of $�$ and $-3�$:

${�}^{7-4}={�}^3$ $(-3�{)}^4=(-3{)}^4{�}^4=81{�}^4$

Combine the results:

${�}_5=35{�}^3\cdot 81{�}^4=2835{�}^3{�}^4$

So, the 5th term in the expansion of $(�-3�{)}^7$ is $2835{�}^3{�}^4$.

In this example, the Binomial Theorem formula for the $(�+1)$-th term was used to find the 5th term in the expansion.