Vardaan
Class 9 Maths • Chapter 2 • NCERT + RS Aggarwal + RD Sharma

Polynomials

Vardaan Learning Institute  |  School-∞Exam Focused Notes

📚 1. Basic Definitions

Polynomial in one variable x: An expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
where aₙ ≠ 0, all powers are non-∞negative integers, and coefficients are real numbers.

NOT polynomials: 1/x (negative power), √x = x^(1/2) (fraction power), 1/(x+1)
Term Definition Example
Degree Highest power of the variable in the polynomial Degree of 3x⁴−2x²+1 is 4
Coefficient Numerical factor of a term In −5x³, coeff. = −5
Constant polynomial Has no variable; degree 0 7, −3, 0
Zero polynomial The polynomial 0; degree undefined (or −∞) p(x) = 0
Linear polynomial Degree 1 ax + b (a ≠ 0)
Quadratic polynomial Degree 2 ax² + bx + c (a ≠ 0)
Cubic polynomial Degree 3 ax³ + bx² + cx + d (a ≠ 0)
Zero/Root of p(x) A value c such that p(c) = 0 If p(x)=x−3, then p(3)=0, zero=3

📌 2. Remainder Theorem

📐 Remainder Theorem (NCERT Theorem 2.3)
If p(x) is a polynomial of degree ≥ 1 and is divided by a linear polynomial (x − a), then the remainder = p(a).

Example: Find remainder when p(x) = x³ − 2x² + 5x − 3 is divided by (x − 2).
Remainder = p(2) = 8 − 8 + 10 − 3 = 7
Extended Remainder Theorem:
When divided by (ax − b), the remainder = p(b/a)
When divided by (ax + b), the remainder = p(−b/a)
Example: Remainder when 3x³ + 7x² − 2x + 5 is divided by (3x − 1):
= p(1/3) = 3(1/27) + 7(1/9) − 2(1/3) + 5 = 1/9 + 7/9 − 6/9 + 45/9 = 47/9

✂️ 3. Factor Theorem

📐 Factor Theorem (NCERT Theorem 2.4)
(x − a) is a factor of p(x) if and only if p(a) = 0.
Equivalently: p(a) = 0 ⟺ (x − a) is a factor of p(x).

Example: Is (x + 1) a factor of p(x) = x³ + x² + x + 1?
p(−1) = −1 + 1 − 1 + 1 = 0 → Yes, (x+1) is a factor ✓
Factorisation using Factor Theorem — Complete Method:
Factorise x³ − 3x² − 10x + 24
Step 1: Try x = 1: 1 − 3 − 10 + 24 = 12 ≠ 0. Not a factor.
Step 2: Try x = 2: 8 − 12 − 20 + 24 = 0 ✓ → (x − 2) is a factor.
Step 3: Divide p(x) by (x − 2): → quotient = x² − x − 12
Step 4: Factorise x² − x − 12 = (x − 4)(x + 3)
∴ x³ − 3x² − 10x + 24 = (x−2)(x−4)(x+3)

🧮 4. Algebraic Identities — Complete List

Identity 1
(a + b)² = a² + 2ab + b²
(x+3)² = x²+6x+9
Identity 2
(a − b)² = a² − 2ab + b²
(x−5)² = x²−10x+25
Identity 3
(a + b)(a − b) = a² − b²
(x+7)(x−7) = x²−49
Identity 4
(x+a)(x+b) = x²+(a+b)x+ab
(x+3)(x+5) = x²+8x+15
Identity 5
(a+b+c)² = a²+b²+c²+2ab+2bc+2ca
(x+y+z)²=x²+y²+z²+2xy+2yz+2zx
Identity 6
(a + b)³ = a³+3a²b+3ab²+b³
(x+2)³ = x³+6x²+12x+8
Identity 7
(a − b)³ = a³−3a²b+3ab²−b³
(x−1)³ = x³−3x²+3x−1
Identity 8 ⭐
a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca)
Special case: if a+b+c=0 then a³+b³+c³=3abc
a³ + b³ = (a+b)(a²−ab+b²)   |   a³ − b³ = (a−b)(a²+ab+b²)
⭐ Board Exam Trick — Identity 8 If a+b+c = 0, then a³+b³+c³ = 3abc.
Example: If x + y + z = 0, find x³ + y³ + z³.
Answer: 3xyz (directly by the identity — no calculation needed!)

📊 5. Long Division of Polynomials

Divide p(x) = 2x³ + 3x² − 8x + 3 by g(x) = x − 2

Step 1: Arrange in descending powers. Divide first term: 2x³ ÷ x = 2x²
Step 2: 2x²(x − 2) = 2x³ − 4x². Subtract: (2x³+3x²) − (2x³−4x²) = 7x²
Step 3: Bring down −8x → 7x² − 8x. Divide: 7x² ÷ x = 7x
Step 4: 7x(x − 2) = 7x² − 14x. Subtract: (7x²−8x) − (7x²−14x) = 6x
Step 5: Bring down 3 → 6x + 3. Divide: 6x ÷ x = 6
Step 6: 6(x − 2) = 6x − 12. Subtract: (6x+3) − (6x−12) = 15
Quotient = 2x² + 7x + 6, Remainder = 15
Verify: p(2) = 16 + 12 − 16 + 3 = 15 ✓ (Remainder Theorem check)
Division Algorithm: p(x) = g(x) × q(x) + r(x)

where degree of r(x) < degree of g(x), or r(x) = 0 (exact division).

✏️ Practice Questions — Polynomials (35 Questions)

Section A — 1 Mark Easy

Q1. Degree of 5x³ − 4x + 7?
Q2. Is x⁻² + 3x + 1 a polynomial? Why?
Q3. Write a quadratic polynomial with zeros 2 and −3.
Q4. Find p(2) if p(x) = x³ − 3x + 1.
Q5. How many zeros can a linear polynomial have?
Q6. Factorise: x² − 5x + 6
Q7. Expand: (2x + 3y)²
Q8. Find the zero of 3x − 5.

Section B — 2/3 Mark Medium

Q9. Find remainder: p(x) = x³ − 6x² + 2x − 4 ÷ (x − 3). Use Remainder Theorem.
Q10. Find remainder when x³ + 3x² + 3x + 1 is divided by (x + 1).
Q11. Is (x − 2) a factor of x³ − 3x + 2. Verify by Factor Theorem.
Q12. Factorise: x² + 5x + 6
Q13. Factorise: 6x² + 17x + 5
Q14. Expand using identity: (2x − 3y + z)²
Q15. Evaluate using identity: 102² = (100+2)²
Q16. Evaluate: 9.8² using identity (10−0.2)²

Section C — 3/4 Mark Medium

  1. Q17. Factorise x³ − 23x² + 142x − 120 using factor theorem. (Hint: try x = 1)
  2. Q18. Factorise: 2x³ + 3x² − 17x + 12
  3. Q19. If p(x) = x³ + ax² + bx + 6 has remainder 3 when divided by (x−3) and (x−2) is a factor, find a and b.
  4. Q20. Expand: (x + 1)(x + 2)(x + 3)
  5. Q21. Evaluate: (2.9)³ = (3−0.1)³ using identity.
  6. Q22. If x + y + z = 6 and xy + yz + zx = 11, find x² + y² + z².
  7. Q23. If a + b + c = 0, prove that a³ + b³ + c³ = 3abc.
  8. Q24. Divide 3x⁴ − 4x³ − 3x − 1 by x − 1 by long division and find quotient and remainder.

Section D — 5 Mark Hard

  1. Q25. If p(x) = x³ − ax² + bx − a has a factor (x − 1), find the relationship between a and b. Then factorise p(x) given a=5, b=7.
  2. Q26. Find a and b if 2x⁴ + ax³ − 14x² + bx + 8 is exactly divisible by x² − 3x + 2.
  3. Q27. Factorise: 27x³ + y³ + z³ − 9xyz
  4. Q28. If x = (1/(√3+√2)) and y = (1/(√3−√2)), find x+y and x−y. Also find x³+y³.
  5. Q29. If (x − 1/x) = 5, find x³ − 1/x³.
  6. Q30. Factorise: x⁶ − y⁶ (Hint: write as (x²)³ − (y²)³ and also as (x³)² − (y³)²)
  7. Q31. Without actual division, show that x⁴ + 4x³ + 4x² − x − 2 has (x+2) as a factor. Factorise completely.
  8. Q32. Evaluate: using a³+b³+c³−3abc identity: if a=2, b=−3, c=4. Also verify by direct calculation.
  9. Q33. If x+y = 5 and xy = 6, find x³+y³ using identity (x+y)³ = x³+y³+3xy(x+y).
  10. Q34. Factorise: 64a³ − 27b³ − 144a²b + 108ab²
  11. Q35. (Challenge) If p(x) = ax³ + bx² + cx + d and p(1) = p(−1), prove that a + c = 0. What does this tell about the polynomial?
✅ Key Answers: Q1:3 | Q3:(x−2)(x+3)=x²+x−6 | Q4:3 | Q6:(x−2)(x−3) | Q8:5/3 | Q9:p(3)=−31 | Q10:0 | Q15:10404 | Q17:(x−1)(x−2)(x−60) | Q22:14 | Q24:Q:3x³−x²−x−2, R:−3 | Q27:(3x+y+z)(9x²+y²+z²−3xy−yz−3xz) | Q29:140 | Q33:x³+y³=35

📝 Quick Revision

  1. Polynomial: only non-∞negative integer powers of variable. 1/x, x^(1/2) are NOT polynomials.
  2. Remainder Theorem: p(x) ÷ (x−a) → remainder = p(a)
  3. Factor Theorem: (x−a) is a factor ⟺ p(a) = 0
  4. 8 key identities — especially (a+b+c)², (a+b)³, (a−b)³, and a³+b³+c³−3abc
  5. If a + b + c = 0 → a³ + b³ + c³ = 3abc (very common board question!)
  6. Division Algorithm: p(x) = g(x)·q(x) + r(x) where deg(r) < deg(g)