Binomial Expansion Worked Examples
The binomial theorem is a powerful tool that allows us to expand expressions of the form $(x + y)^n$ for any positive integer $n$. It states that:
$(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$
where $\binom{n}{k}$ is the binomial coefficient, which represents the number of ways to choose $k$ objects from a set of $n$ objects. It is calculated as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Let's look at some worked examples to understand how the binomial theorem is applied.
Example 1: Expanding $(x + y)^3$
Using the binomial theorem, we have:
$(x + y)^3 = \binom{3}{0}x^3y^0 + \binom{3}{1}x^2y^1 + \binom{3}{2}x^1y^2 + \binom{3}{3}x^0y^3$
Now, we calculate the binomial coefficients:
- $\binom{3}{0} = \frac{3!}{0!3!} = 1$
- $\binom{3}{1} = \frac{3!}{1!2!} = 3$
- $\binom{3}{2} = \frac{3!}{2!1!} = 3$
- $\binom{3}{3} = \frac{3!}{3!0!} = 1$
Substituting these values back into the equation:
$(x + y)^3 = 1x^3 + 3x^2y + 3xy^2 + 1y^3$
Therefore, the expanded form of $(x + y)^3$ is $\boxed{x^3 + 3x^2y + 3xy^2 + y^3}$.
Example 2: Expanding $(2a - b)^4$
We can rewrite the expression as $(2a + (-b))^4$ and apply the binomial theorem:
$(2a - b)^4 = \binom{4}{0}(2a)^4(-b)^0 + \binom{4}{1}(2a)^3(-b)^1 + \binom{4}{2}(2a)^2(-b)^2 + \binom{4}{3}(2a)^1(-b)^3 + \binom{4}{4}(2a)^0(-b)^4$
Calculating the binomial coefficients:
- $\binom{4}{0} = 1$
- $\binom{4}{1} = 4$
- $\binom{4}{2} = 6$
- $\binom{4}{3} = 4$
- $\binom{4}{4} = 1$
Substituting these values:
$(2a - b)^4 = 1(16a^4) + 4(8a^3)(-b) + 6(4a^2)(b^2) + 4(2a)(-b^3) + 1(b^4)$
Therefore, the expanded form of $(2a - b)^4$ is $\boxed{16a^4 - 32a^3b + 24a^2b^2 - 8ab^3 + b^4}$.
Example 3: Finding a specific term in the expansion
Let's find the term containing $x^5$ in the expansion of $(2x - 3)^8$.
Using the binomial theorem, the general term in the expansion is:
$\binom{8}{k}(2x)^{8-k}(-3)^k$
We want the term where $8-k = 5$, which means $k=3$.
Substituting $k = 3$ into the general term:
$\binom{8}{3}(2x)^5(-3)^3 = 56(32x^5)(-27) = \boxed{-48384x^5}$
Therefore, the term containing $x^5$ in the expansion of $(2x - 3)^8$ is $-48384x^5$.
These examples demonstrate the application of the binomial theorem in expanding binomial expressions and finding specific terms within those expansions. Remember to pay attention to the signs and coefficients when applying the theorem, and practice will help you become more comfortable with the process.
