1st Year Math Chapter 8 Exercise 8.2 Notes: Binomial Theorem for a Positive Integer Index (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Exercise 8.2 introduces the Binomial Theorem, which gives a direct formula for expanding (a + x)ⁿ for any positive integer n, using the combination formulas from Chapter 7.
What Does This Exercise Cover?
This exercise teaches how to expand a binomial raised to a power without multiplying it out term by term, how to find any specific term in the expansion, and how to locate the middle term.
Key Concepts
1. The Binomial Theorem
For a positive integer n, (a + x)ⁿ expands as the sum of terms nC0 aⁿ + nC1 aⁿ⁻¹x + nC2 aⁿ⁻²x² + … + nCn xⁿ, where the coefficients nCr are the combination values from Chapter 7.
2. The General Term
The (r+1)th term of the expansion, called the general term, is given by Tᵣ₊₁ = nCr × aⁿ⁻ʳ × xʳ. This formula lets you find any specific term without expanding the whole expression.
3. The Middle Term
If n is even, the expansion has exactly one middle term, at position (n/2 + 1). If n is odd, there are two middle terms, at positions (n+1)/2 and (n+3)/2.
4. Deductions from the Binomial Expansion
Substituting a = 1 and x = 1 shows that the sum of all binomial coefficients equals 2ⁿ. Substituting a = 1 and x = −1 shows that the alternating sum of the coefficients equals 0 (for n ≥ 1).
Solved Examples
Example 1: Expand (a + x)⁴ using the binomial theorem.
(a+x)⁴ = 4C0 a⁴ + 4C1 a³x + 4C2 a²x² + 4C3 ax³ + 4C4 x⁴.
Substituting the combination values: 1, 4, 6, 4, 1.
Answer: a⁴ + 4a³x + 6a²x² + 4ax³ + x⁴
Example 2: Find the 4th term in the expansion of (a + x)⁷.
The 4th term corresponds to r = 3 (since Tᵣ₊₁ is the 4th term when r = 3).
T₄ = 7C3 × a^(7−3) × x³ = 35 a⁴x³.
Answer: 35a⁴x³
Example 3: Find the middle term of (a + x)⁶.
n = 6 is even, so the middle term is at position (6/2 + 1) = 4.
T₄ = 6C3 × a³ × x³ = 20a³x³.
Answer: 20a³x³
Sample MCQs
1. The binomial theorem gives the expansion of:
a) (a + x)ⁿ b) a + x = n c) aⁿ + xⁿ d) a × xⁿ
Answer: a) (a + x)ⁿ
2. The general term in the expansion of (a + x)ⁿ is:
a) nCr aʳ x^(n−r) b) nCr a^(n−r) xʳ c) nPr a^(n−r) xʳ d) n! aⁿ xⁿ
Answer: b) nCr a^(n−r) xʳ
3. If n is even, the expansion of (a + x)ⁿ has:
a) Two middle terms b) One middle term c) No middle term d) n middle terms
Answer: b) One middle term
4. Setting a = x = 1 in the binomial expansion of (a + x)ⁿ gives:
a) n b) 2ⁿ c) n! d) 0
Answer: b) 2ⁿ
5. The 3rd term (r = 2) in the expansion of (a + x)⁵ is:
a) 5C2 a³x² b) 5C2 a²x³ c) 5C3 a²x³ d) 5C1 a⁴x
Answer: a) 5C2 a³x²
Important Short Questions
- State the binomial theorem for (a + x)ⁿ where n is a positive integer.
- Write the general term formula for the expansion of (a + x)ⁿ.
- Expand (a + x)³ completely.
- Find the number of terms in the expansion of (a + x)⁸.
- Explain how to determine whether an expansion has one or two middle terms.
Important Long Questions
- Expand (a + x)⁵ completely using the binomial theorem.
- Find the 5th term in the expansion of (2x + 3)⁸, without expanding the full expression.
- Find the middle term(s) in the expansion of (a + x)⁷.
- Show that the sum of the binomial coefficients in the expansion of (a + x)ⁿ equals 2ⁿ, using a = x = 1.
How to Approach This Exercise Effectively
- Remember the expansion of (a+x)ⁿ always has exactly n + 1 terms — use this to double-check you haven’t missed one.
- For ‘find the kth term’ questions, be careful with the off-by-one relationship: the kth term corresponds to r = k − 1 in the general term formula.
- Determine whether n is even or odd immediately when asked for a middle term — this tells you whether to expect one term or two.
- Reuse your combination (nCr) skills from Chapter 7 directly here — the coefficients in a binomial expansion are exactly those values.
- Practice both fully expanding small binomials and finding just a specific term of a larger one, since exams test both skills separately.
FAQs
Q: What does ‘nCr’ represent in the binomial expansion?
A: It’s the combination value from Chapter 7, giving the coefficient of each term in the expansion — specifically, the coefficient of the term where x is raised to the power r.
Q: How many terms are there in the expansion of (a + x)ⁿ?
A: There are always exactly n + 1 terms, since r ranges from 0 to n.
Q: Why does an odd value of n give two middle terms instead of one?
A: With n + 1 total terms, an odd n means an even total term count, which has no single middle position, so two terms share the ‘middle’ of the expansion.
Q: What is the binomial theorem used for beyond expanding expressions?
A: It’s used to quickly find specific terms or coefficients in large expansions, to derive combinatorial identities, and it forms the basis for the extended binomial series studied in Exercise 8.3.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 8: Mathematical Induction and Binomial Theorem, Exercise 8.2.