Binomial Theorem | Class 11 Mathematics Notes

7.2 Binomial Theorem for Positive Integral Indices

This section formally introduces the Binomial Theorem for positive integral indices by revisiting known expansions of binomials raised to powers 0 through 4. The expansions are:

(a + b)^0 = 1 (for a + b ≠ 0) (a + b)^1 = a + b (a + b)^2 = a² + 2ab + b² (a + b)^3 = a³ + 3a²b + 3ab² + b³ (a + b)^4 = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

From these expansions, several important observations are made:

(i) The number of terms in the expansion is always one more than the power n. For example, (a + b)^2 has 3 terms.

(ii) The powers of the first term 'a' decrease by 1 in successive terms, starting from n down to 0.

(iii) The powers of the second term 'b' increase by 1 in successive terms, starting from 0 up to n.

(iv) In each term, the sum of the exponents of 'a' and 'b' equals the power n.

These observations set the stage for understanding the coefficients that appear in the expansion, which are later linked to binomial coefficients and Pascal's triangle. The section also introduces the arrangement of coefficients in a tabular form, showing the pattern of numbers that appear in the expansions for increasing powers. This pattern is foundational for the development of the Binomial Theorem.

📊 Diagram: See table_1: Table showing coefficients for powers 0 to 4

🧪 Activity: No specific activity, but encourages observation of coefficient patterns.

🔗 Connection: Leads to the introduction of Pascal's triangle as a tool to find binomial coefficients.

Table on page 2 (6×6)