BINOMIAL THEOREM

Definition Of Binomial Theorem

Binomial Theorem is used to expand powers of binomials.
It States that :
(a + b)ⁿ = _nC₀ aⁿ + _nC₁ a^n-1b¹ + _nC₂ a^n-2b² +..........+ _nC_n-1 ab^n-1 + _nC_nbⁿ,
where n = 0
Binomial Theorem also finds the coefficient of a term in the expansion.

Examples of Binomial Theorem

(x + 4)⁴ = x⁴ + 16x³ + 96x² + 256x + 256
The power of the binomial x + 4 is distributed using the binomial theorem.

Video Examples: Binomial Theorem

Solved Example on Binomial Theorem

Ques: Find the coefficient of q³ in the expansion of (q - 1)³.

Choices:

A. 1
B. 0
C. 3
D. -1
Correct Answer: A

Solution:

Step 1: Expanding (q - 1)³ by using the binomial theorem, we get (q - 1)³ = q³ - 3q² + 3q - 1
Step 2: The coefficient of q³ in the above expansion is 1.

Quick Summary

Expands binomials raised to a power.
Finds specific coefficients in an expansion.
Uses combinations (nCr) to determine coefficients.

\[ (a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k \]

🍎 Teacher Insights

Emphasize the pattern of coefficients and exponents. Use Pascal's Triangle as a visual aid. Provide ample practice with different values of 'n' and binomial expressions.

🎓 Prerequisites

Algebraic expansion
Combinations
Exponents

Check Your Knowledge

Q1: What is the coefficient of x² in the expansion of (x + 1)³?

1 2 3 4

Q2: What is the last term in the expansion of (a + b)⁴?

4a b⁴ 4b⁴ a⁴

Frequently Asked Questions

Q: What is a binomial?
A: An expression with two terms, like (x + y).

Q: How does the binomial theorem work?
A: It provides a formula for expanding (a + b)^n into a sum of terms.