Binomial expansion

Definition

Binomial expansion is the process of expanding an expression that is raised to a power, specifically in the form $(a + b)^n$. It utilizes the Binomial Theorem to provide a series of terms involving coefficients, powers of $a$, and powers of $b$.

5 Must Know Facts For Your Next Test

1. The general term in the binomial expansion of $(a + b)^n$ is given by $T_k = \binom{n}{k} a^{n-k} b^k$.
2. The coefficients in the expansion are binomial coefficients, which can be found using Pascal's Triangle or the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
3. The sum of the exponents in each term of the expanded form is always equal to $n$.
4. If both $a$ and $b$ are set to 1, the sum of all coefficients in $(a + b)^n$ equals $2^n$.
5. The number of terms in the expanded form is always $n+1$, where $n$ is the exponent.

Review Questions

• What is the coefficient of the term containing $a^3b^2$ in the expansion of $(a + b)^5$?
• How do you find the general term in a binomial expansion?
• What pattern do binomial coefficients follow that helps simplify their calculation?

Related terms

Binomial Coefficient: A numerical factor that multiplies each term in a binomial expansion, represented as $\binom{n}{k}$.

Pascal's Triangle: A triangular array where each entry is a binomial coefficient, used to find coefficients for binomial expansions.

$(a+b)^2$ Formula: $(a+b)^2 = a^2 + 2ab + b^2$, illustrating how binomials expand at lower powers.