PRE-CALCULUS: UNIT 8.4

EXPANSION
BLUEPRINT

Combinatorics meets Binomial Expansion

SYSTEMS READY

WARM-UP

5:00 MINUTES

Expand the following expressions in your notebook:

01 \[ (x + y)^2 \]

02 \[ (x + y)^3 \]

Show every step of your distribution or FOIL process.

Reviewing Combinations

Permutations vs. Combinations Review

Key Formula

\[ C(n,r) = \frac{n!}{(n-r)!r!} \]

Order Matters?

No. We divide by \(r!\) to remove duplicate arrangements.

COMBINATORICS

THE HIDDEN PATTERN

Algebraic Expansion

POWER: n = 2

\[ 1x^2 + \mathbf{2}xy + 1y^2 \]

POWER: n = 3

\[ 1x^3 + \mathbf{3}x^2y + \mathbf{3}xy^2 + 1y^3 \]

Combinatorial Logic

C(2,k)

\[ C(2,0), C(2,1), C(2,2) \]

C(3,k)

\[ C(3,0), C(3,1), C(3,2), C(3,3) \]

The coefficients are exactly the values in Pascal's Triangle!

THE BINOMIAL THEOREM

The general formula for any positive integer \(n\):

\[ (x + y)^n = \sum_{k=0}^{n} C(n,k) x^{n-k} y^k \]

\(C(n,k)\)

The "Binomial Coefficient"

\(x^{n-k}\)

Power of \(x\) decreases

\(y^k\)

Power of \(y\) increases

ACTIVITY: THE BLUEPRINT REWRITE

Task Instructions

Return to your Warm-up expansions.
Replace every numerical coefficient with its Combination notation.
Calculate the combination to prove it matches your expansion.

Pro-Tip

Remember that the first term's index is always \(k=0\). If the power is \(n=3\), there will be \(n+1\) (four) terms.

Example Structure

\( (x+y)^2 = \) \( \mathbf{C(2,0)}x^2 + \mathbf{C(2,1)}xy + \mathbf{C(2,2)}y^2 \)

\( (x+y)^3 = \) \( \mathbf{C(3,0)}x^3 + \dots \)

FINAL CHALLENGE

PREDICTION TIME

Without doing any multiplication...

What are the coefficients for \((x + y)^6\)?

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Write your predicted combination notation list now!

Expansion Blueprint

Expansion Blueprint Lesson Plan

Expansion Blueprint Whiteboard Slides

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