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

Polynomials

Vardaan Learning Institute  |  School-8Exam Focused Notes

?? 1. Basic Definitions

Polynomial in one variable x: An expression of the form a?xn + a??1xn?¹ + ... + a1x + a0
where a? ? 0, all powers are non-8negative integers, and coefficients are real numbers.

NOT polynomials: 1/x (negative power), vx = 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 -8) 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/(v3+v2)) and y = (1/(v3-v2)), 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-8negative 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)