Interactive Binomial Expansion

1. The General Formula

In basic math, you learned $(a+b)^n$ for integers (Pascal's Triangle). But what if $n$ is a fraction or negative?

The Formula (for $|x| < 1$): $$ (1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots $$

Quick Check: Does this formula work for $(1+x)^{-2}$?

2. Identifying $n$ and $x$

The formula is strictly for $(1 + \mathbf{something})^\mathbf{power}$.

Look at this expression: $$ (1 - 4x)^{\frac{1}{2}} $$

Fill in the blanks to prepare for expansion:

$n$ (The Power) =

Be careful with the sign!

$x$ (The term replaced) =

Correct! When we expand, every time we see "$x$" in the formula, we will write $(-4x)$.

3. Let's Calculate Terms

Expand $(1 - x)^{-2}$ up to $x^2$.

Here, $n = -2$ and replace $x$ with $(-x)$.

1. First Term: Always starts with: 1

2. Second Term ($nx$): $$ n \cdot x \rightarrow (-2) \cdot (-x) $$ Coefficient is: $x$

3. Third Term ($\frac{n(n-1)}{2!}x^2$):
Calculate $n(n-1)$: $(-2)(-3) = 6$.
Divide by $2!$ (which is 2): $3$.
Multiply by replacement $x^2$: $(-x)^2 = x^2$.
Resulting Coefficient: $x^2$

🎉 Result

$$ (1-x)^{-2} = 1 + 2x + 3x^2 + \dots $$

Notice the pattern? For negative integer powers, we often get whole number coefficients.

4. Validity (Convergence)

This expansion represents an infinite series. It only "works" (sums to a real answer) if the $x$-term is small.

Rule: $| \text{term containing } x | < 1$

Example: For $(1 + 3x)^{-1}$

Condition: $|3x| < 1$

Rearrange $|3x| < 1$ to find the range of $x$.

$$ |x| < \dots $$ Input the boundary:

Correct! The expansion is only valid when $x$ is between $-\frac{1}{3}$ and $\frac{1}{3}$.

5. Factorization Trick

What if the first number isn't 1?
Example: $(4 + x)^{\frac{1}{2}}$

You cannot use the formula directly! You must force the 1.

Step 1: Factor out the 4 inside the bracket $$ ( 4(1 + \frac{x}{4}) )^{\frac{1}{2}} $$

Step 2: Apply the power to the 4 $$ 4^{\frac{1}{2}} (1 + \frac{x}{4})^{\frac{1}{2}} $$

We know $4^{1/2} = \sqrt{4} = 2$.

Step 3: Setup the Expansion $$ 2 \times [ \text{Expand } (1 + \frac{x}{4})^{\frac{1}{2}} ] $$ Here, $x$ is replaced by $\frac{x}{4}$.