The binomial expansion

What this chapter teaches

The binomial theorem writes (a + b)ⁿ as a sum of terms nCr a^(n − r) b^r, for a positive whole number n. A single term can be found without the rest. For a negative or fractional power, (1 + x)ⁿ becomes an endless series 1 + nx + n(n − 1)/2! x² + …, and it gives the right value only when x is small: |x| < 1 (or |bx| < 1 for (1 + bx)ⁿ).

Three questions to ask

There is no one right wording. Listen for the idea.

1. What is the coefficient of x in (2 + x)⁵, and where does it come from?
2. Why does the expansion of (1 + x)⁻¹ not stop, while (1 + x)³ does?
3. For which values of x is 1 − x + x² − x³ + … a good value of 1/(1 + x)?

After the chapter, the learner can

• Expand (2 + x)⁵ with the binomial theorem
• Find one term, like the x³ term, without the full expansion
• Expand (1 + x)⁻¹ for small x and say when it is valid