Lesson 13.6: Binomial Theorem

New Concept

The Binomial Theorem offers a powerful formula to expand binomials like $(x+y)^n$ without tedious multiplication. You'll learn to apply it by identifying patterns in exponents and calculating binomial coefficients to find the complete expansion or a single term.

What’s next

Next, you'll tackle interactive examples expanding binomials and solve challenge problems to find specific terms using the powerful Binomial Theorem formula.

Property

If $n$ and $r$ are integers greater than or equal to 0 with $n \geq r$, then the binomial coefficient is

An important property of binomial coefficients is that they are symmetric, meaning:

Examples

• Find the binomial coefficient $\binom{6}{2}$: $\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \cdot 5}{2 \cdot 1} = 15$.

• Find the binomial coefficient $\binom{8}{3}$: $\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 56$.

Property

The Binomial Theorem is a formula that can be used to expand any binomial.

To expand a binomial:

1. Determine the exponent, $n$.
2. Use the formula to evaluate each term from $k=0$ to $k=n$.
3. Simplify the result.

Examples

• Expand $(a+b)^3$: This becomes $\binom{3}{0}a^3b^0 + \binom{3}{1}a^2b^1 + \binom{3}{2}a^1b^2 + \binom{3}{3}a^0b^3$, which simplifies to $a^3 + 3a^2b + 3ab^2 + b^3$.

• Expand $(2x-1)^4$: Using the theorem, we get $1(2x)^4 + 4(2x)^3(-1) + 6(2x)^2(-1)^2 + 4(2x)(-1)^3 + 1(-1)^4$, which simplifies to $16x^4 - 32x^3 + 24x^2 - 8x + 1$.

Property

The $(r + 1)$th term of the binomial expansion of $(x + y)^n$ is:

To find a specific term without fully expanding:

1. Determine the exponent, $n$.
2. Identify which term you need, $(r+1)$.
3. Calculate $r$ from $r+1$.
4. Substitute $n$ and $r$ into the formula.

Examples

• Find the third term of $(a+b)^{10}$. Here, $n=10$ and $r+1=3$, so $r=2$. The term is $\binom{10}{2}a^{10-2}b^2 = 45a^8b^2$.

• Find the fourth term of $(2x-y)^7$. Here, $n=7$ and $r+1=4$, so $r=3$. The term is $\binom{7}{3}(2x)^{7-3}(-y)^3 = 35(16x^4)(-y^3) = -560x^4y^3$.