The Binomial Theorem provides a formula for expanding expressions of the form $(x + y)^n$, where $n$ is a non-negative integer. Instead of multiplying $(x + y)$ by itself $n$ times, the theorem uses combinations (represented by $nCr$ or $\binom{n}{r}$) to find the coefficients of each term in the expansion. Each term has the form $\binom{n}{r}x^{n-r}y^r$, where $r$ ranges from $0$ to $n$. This makes expanding binomials with high powers much simpler and faster!
In essence, the Binomial Theorem states:
$(x + y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} y^r$
Match each term with its definition:
| Term | Definition |
|---|---|
| 1. Binomial | a. A number that multiplies a variable. |
| 2. Coefficient | b. The power to which a variable is raised. |
| 3. Exponent | c. An expression with two terms. |
| 4. Combination | d. A way to select items from a set where order doesn't matter. |
| 5. Factorial | e. The product of all positive integers less than or equal to a given number. |
The Binomial Theorem uses ______ to expand expressions like $(x + y)^n$. The general term in the expansion is given by $\binom{n}{r}x^{n-r}y^r$, where $\binom{n}{r}$ represents the ______ coefficient, calculated as $\frac{n!}{r!(n-r)!}$. The symbol '!' denotes a ______. When expanding, the exponents of $x$ ______ while the exponents of $y$ increase. The sum of the exponents in each term always equals ______.
Explain in your own words why using the Binomial Theorem is more efficient than directly multiplying $(x + y)$ by itself $n$ times, especially when $n$ is a large number. Provide an example.
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