College Algebra / Unit 9 Alg 522 worked example, 60 to 90 seconds
Expanding with the Binomial Theorem
The idea
Expanding with the Binomial Theorem means writing the coefficient row, walking one base's exponent down while the other's climbs, and simplifying each term.
In practice
Expand (x + 2)³. The row for n = 3: 1, 3, 3, 1. The powers of x fall 3, 2, 1, 0 while the powers of 2 rise 0, 1, 2, 3. Assemble: x³ + 3x²(2) + 3x(4) + 8, which simplifies to x³ + 6x² + 12x + 8. The audit is one substitution: at x = 1, the original is 3³ = 27, and the expansion answers 1 + 6 + 12 + 8 = 27. Exponents in every term sum to n — a second free check — and the term count is n + 1, never n. Any binomial expands this way, subtraction included: a − b runs the same row with alternating signs.
Where people slip
Raise the whole second base to its power; the 2³ is 8, not 6.
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
Builds on
- Nothing. This is a starting point.
Unlocks
- Nothing yet depends on this.
Something wrong in this lesson? Flag it · Sources for College Algebra