ACT Math: Binomial Theorem—Expand (a+b)^n Without Multiplying

The binomial theorem states: (a+b)^n=Σ C(n,k)a^(n-k)b^k for k=0 to n, where C(n,k) is the binomial coefficient (n choose k). In practice, use Pascal's triangle to find coefficients: Row 0: 1. Row 1: 1 1. Row 2: 1 2 1. Row 3: 1 3 3 1. Row 4: 1 4 6 4 1. Each row gives the coefficients for (a+b)^n where n is the row number. For example, (a+b)^3=1a³+3a²b+3ab²+1b³ (using row 3: 1, 3, 3, 1).

Advantage over manual expansion: No need to multiply (a+b)(a+b)(a+b)... ; just read off coefficients and powers from Pascal's triangle.

Example 1: (x+2)²=1x²+2(x)(2)+1(2)²=x²+4x+4. (Coefficients 1,2,1 from row 2.) Example 2: (a+b)⁴. Row 4: 1,4,6,4,1. (a+b)⁴=a⁴+4a³b+6a²b²+4ab³+b⁴. Example 3: (2x-3)³. Row 3: 1,3,3,1. With a=2x, b=-3: (2x)³+3(2x)²(-3)+3(2x)(-3)²+(-3)³=8x³-36x²+54x-27. Key: powers of a decrease from n to 0; powers of b increase from 0 to n. Coefficients come from Pascal's triangle.

Practice: Expand (x+1)^5 using row 5 (1,5,10,10,5,1).

Expansion 1: (a+b)^3=a³+3a²b+3ab²+b³. Expansion 2: (2x+1)²=4x²+4x+1. Check: coefficients are 1,2,1 from row 2. (2x)²=4x², 2(2x)(1)=4x, 1²=1. ✓ Expansion 3: (x-y)⁴=x⁴-4x³y+6x²y²-4xy³+y⁴. (Row 4: 1,4,6,4,1. b=-y makes terms alternate signs.) Complete all three daily until you expand binomials quickly using Pascal's triangle.

Verify one expansion by manual multiplication to confirm Pascal's triangle gives the right coefficients.

Binomial expansion questions appear in 1-2 ACT Math sections, usually asking you to expand a binomial or find a specific term. Using Pascal's triangle is much faster than multiplying manually. A student who knows the binomial theorem solves these in 1-2 minutes; one who multiplies manually takes 5+ minutes. That time saved is buffer for harder questions.

Master Pascal's triangle and the binomial theorem in one study session. By test day, expanding binomials becomes quick.