38. The binomial theorem

The lesson, step by step

1. Warm-up

   Ask your student to multiply out (a + b)³ the long way.

2. Watch the pattern

   Read Art. 183 and look at the powers of a + b up to the fifth. Ask: What do you notice about the number of terms, the exponents, the coefficients?

3. The four laws

   Read Art. 184.

   The number of terms in any power of a binomial is one greater than the exponent of the power.

   Then the law of coefficients:

   If the coefficient of any term be multiplied by the exponent of the leading letter, and the product be divided by the number of that term from the left, the quotient will be the coefficient of the next term.

4. Work the sixth power together

   Coefficients: 1; 1 × 6 ÷ 1 = 6; 6 × 5 ÷ 2 = 15; 15 × 4 ÷ 3 = 20; then they repeat: 15, 6, 1. So (a + b)⁶ = a⁶ + 6a⁵b + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6ab⁵ + b⁶.

5. Guided practice

   Your student writes (x − y)⁴ and (m + n)⁵ (exercises 2 and 3) while you watch. For the minus sign, the even-numbered terms are negative.

6. Compound terms

   Read Art. 185 and do exercise 2 together:

   2. Find the cube of 2a— 3b.

   Let m = 2a and n = 3b; (m − n)³ = m³ − 3m²n + 3mn² − n³ = 8a³ − 36a²b + 54ab² − 27b³.

7. Independent practice

   Exercises 4 to 8 of Art. 184 and 3 to 5 of Art. 185. After working by hand, your student can confirm one on the site's calculator with expand((m+2*n)^4).

8. Wrap-up

   Ask: Without writing it, how many terms does (a + b)¹⁰ have? What is its second term? (11; 10a⁹b.)