AP PRECALCULUS — UNIT 1B · POLYNOMIAL & RATIONAL FUNCTIONS
1.11A — Equivalent Expressions and the Binomial Theorem
Notes — Expanding Powers of Binomials
💡 Learning Objectives
By the end of this lesson you will be able to (AP CED 1.11.A):
- Expand (a + b)ⁿ for small positive integer n using Pascal's Triangle.
- Apply the Binomial Theorem to find a specific term in an expansion.
- Rewrite polynomial expressions in equivalent factored or expanded forms.
- Use equivalent forms to reveal specific features (zeros, leading coefficient, constant term).
1. Why Equivalent Forms Matter
A polynomial can be written in many equivalent forms — expanded, factored, or as a product. Each form reveals something different. Factored form shows zeros. Expanded form shows the leading coefficient and the constant term. Moving fluently between forms is an essential AP precalculus skill.
💡 Quick Check
Identify what each form of f(x) = x² − 5x + 6 reveals.
- Expanded form x² − 5x + 6: reveals leading coefficient 1 and y-intercept 6.
- Factored form (x − 2)(x − 3): reveals zeros at x = 2 and x = 3.
- Vertex form (x − 5/2)² − 1/4: reveals minimum at (5/2, −1/4).
2. Pascal's Triangle
Pascal's Triangle gives the coefficients of (a + b)ⁿ. Each row starts and ends with 1; every interior entry is the sum of the two entries above it.
Pascal's Triangle — row n gives the binomial coefficients of (a + b)ⁿ.
3. The Binomial Theorem
💡 Definitions
Binomial Theorem. For a positive integer n:
(a + b)ⁿ = C(n,0)·aⁿ + C(n,1)·aⁿ⁻¹·b + C(n,2)·aⁿ⁻²·b² + … + C(n,n−1)·a·bⁿ⁻¹ + C(n,n)·bⁿ
where C(n, k) = n! / (k!·(n − k)!) is the binomial coefficient. These are exactly the numbers in row n of Pascal's Triangle.
📘 Example
Example 1 — Expanding (x + 2)⁴
Row 4 of Pascal's Triangle: 1, 4, 6, 4, 1.
So (x + 2)⁴ = 1·x⁴ + 4·x³·2 + 6·x²·4 + 4·x·8 + 1·16
= x⁴ + 8x³ + 24x² + 32x + 16.
📘 Example
Example 2 — Expanding (3x − 1)³
Treat it as (a + b)³ with a = 3x and b = −1. Row 3 coefficients: 1, 3, 3, 1.
(3x − 1)³ = (3x)³ + 3(3x)²(−1) + 3(3x)(−1)² + (−1)³
= 27x³ − 27x² + 9x − 1.
⚠️ Common Mistake
When b is negative, do NOT drop the minus sign during expansion. The sign alternates from term to term: (+), (−), (+), (−), …. Keep b = −1 (or whatever it is) intact and let the algebra do its work.
4. Finding a Specific Term
The general term of (a + b)ⁿ is C(n, k) · aⁿ⁻ᵏ · bᵏ. This formula lets you jump directly to any term without expanding everything.
📘 Example
Example 3 — The x²-term of (x + 3)⁵
We want the x² term, so n − k = 2 and therefore k = 3.
Term = C(5, 3) · x² · 3³ = 10 · x² · 27 = 270x².
🎯 AP Tip
When an AP question asks for the coefficient of a specific power of x in an expansion, skip to the general term formula — don't expand the whole expression.
📘 Try This
Expand (2x − 1)⁴ using Pascal's Triangle (row 4: 1, 4, 6, 4, 1).
- (2x)⁴ + 4(2x)³(−1) + 6(2x)²(−1)² + 4(2x)(−1)³ + (−1)⁴
- = 16x⁴ − 32x³ + 24x² − 8x + 1.
5. Summary
- Pascal's Triangle gives the binomial coefficients; row n has n + 1 entries.
- The Binomial Theorem: (a + b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ, k from 0 to n.
- The general term C(n, k) · aⁿ⁻ᵏ · bᵏ lets you extract any individual term.
- Equivalent forms reveal different features — factored for zeros, expanded for leading coefficient.