1st Year Math Chapter 9 Exercise 9.1 Notes: Polynomial Division, Remainder Theorem, and Factor Theorem (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.
Chapter 9, Division of Polynomials, follows Mathematical Induction and Binomial Theorem in the 14-unit Mathematics 11 (PECTAA) textbook. Exercise 9.1 covers dividing polynomials by long division, and introduces the Remainder Theorem and the Factor Theorem.
What Does This Exercise Cover?
This exercise teaches how to divide one polynomial by another using long division, and two powerful shortcuts — the Remainder and Factor Theorems — that let you find remainders and test for factors without doing full division.
Key Concepts
1. Dividing Polynomials by Long Division
Dividing a polynomial P(x) by another polynomial D(x) follows the same repeated divide-multiply-subtract process as long division of numbers, producing a quotient Q(x) and a remainder R(x), related by P(x) = D(x) × Q(x) + R(x).
2. The Remainder Theorem
If a polynomial P(x) is divided by (x − c), the remainder is exactly P(c) — the value of the polynomial evaluated at x = c. This lets you find a remainder by simple substitution instead of full division.
3. The Factor Theorem
(x − c) is a factor of P(x) if and only if P(c) = 0. This connects finding roots of a polynomial directly to finding its factors.
Solved Examples
Example 1: Divide x³ + 2x² − 5x − 6 by x − 2 using long division.
x³ ÷ x = x². Multiply: x²(x−2) = x³ − 2x². Subtract: 4x² − 5x − 6.
4x² ÷ x = 4x. Multiply: 4x(x−2) = 4x² − 8x. Subtract: 3x − 6.
3x ÷ x = 3. Multiply: 3(x−2) = 3x − 6. Subtract: 0.
Answer: Quotient: x² + 4x + 3, Remainder: 0
Example 2: Use the Remainder Theorem to find the remainder when P(x) = x³ − 4x² + 3x + 7 is divided by x − 1.
By the Remainder Theorem, the remainder equals P(1).
P(1) = 1 − 4 + 3 + 7.
Answer: Remainder = 7
Example 3: Determine whether (x + 2) is a factor of P(x) = x³ + 2x² − x − 2.
By the Factor Theorem, (x + 2) is a factor if P(−2) = 0.
P(−2) = (−2)³ + 2(−2)² − (−2) − 2 = −8 + 8 + 2 − 2.
Answer: P(−2) = 0, so (x + 2) is a factor
Sample MCQs
1. The relationship between dividend, divisor, quotient, and remainder for polynomials is:
a) P(x) = (x−c)Q(x) + R b) P(x) = (x−c) + Q(x) c) P(x) = Q(x) / (x−c) d) P(x) = R / (x−c)
Answer: a) P(x) = (x−c)Q(x) + R
2. The Remainder Theorem states that dividing P(x) by (x − c) leaves a remainder equal to:
a) 0 b) P(c) c) c d) P(0)
Answer: b) P(c)
3. According to the Factor Theorem, (x − c) is a factor of P(x) if and only if:
a) P(c) = 1 b) P(c) = 0 c) P(0) = c d) c = 0
Answer: b) P(c) = 0
4. If P(3) = 0 for a polynomial P(x), then a factor of P(x) is:
a) (x + 3) b) (x − 3) c) x d) 3x
Answer: b) (x − 3)
5. The remainder when P(x) = x² + 3x + 2 is divided by x − 1 is:
a) 0 b) 4 c) 6 d) 2
Answer: c) 6
Important Short Questions
- State the relationship between dividend, divisor, quotient, and remainder for polynomial division.
- State the Remainder Theorem.
- State the Factor Theorem.
- Find the remainder when P(x) = x³ − 2x + 5 is divided by x − 2, using the Remainder Theorem.
- Determine whether (x − 1) is a factor of P(x) = x³ − 1.
Important Long Questions
- Divide x³ − 3x² + 5x − 3 by x − 1 using long division, identifying the quotient and remainder.
- Use the Remainder Theorem to find the remainder when P(x) = 2x³ − 5x² + 4x − 3 is divided by x + 1.
- Determine whether (x − 3) is a factor of P(x) = x³ − 4x² + x + 6, using the Factor Theorem.
- Explain the connection between the Remainder Theorem and the Factor Theorem.
How to Approach This Exercise Effectively
- Always double-check the sign of c when applying the Remainder or Factor Theorem — dividing by (x + 2) means c = −2, not 2.
- Practice long division carefully, keeping terms aligned by degree, since a missing term (like no x² term) is a common source of mistakes.
- Use the Remainder Theorem as a fast way to check a long division answer — the remainder you compute should match P(c).
- Remember the Factor Theorem is really a special case of the Remainder Theorem, where the remainder happens to be exactly zero.
- When testing several possible factors, evaluate P(c) for each candidate quickly before committing to a full long division.
FAQs
Q: What’s the difference between the Remainder Theorem and the Factor Theorem?
A: The Remainder Theorem finds the remainder for any divisor (x−c); the Factor Theorem is the special case where that remainder equals zero, confirming (x−c) is an exact factor.
Q: Can the Remainder Theorem be used for divisors other than (x − c)?
A: No — it applies specifically to linear divisors of the form (x − c); other divisors require full long division or different techniques.
Q: What does it mean if the remainder is zero?
A: It means the divisor divides the polynomial exactly, with no leftover term — equivalently, the divisor is a factor of the polynomial.
Q: Why is long division of polynomials similar to long division of numbers?
A: Both follow the same repeated cycle of dividing the leading term, multiplying, and subtracting, just applied to polynomial terms instead of digits.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 9: Division of Polynomials, Exercise 9.1.