Algebraic Identities

Standard formulas that are true for all values of variables.

Standard Identities

  • (x + y)² = x² + 2xy + y²
  • (x - y)² = x² - 2xy + y²
  • x² - y² = (x + y)(x - y)
  • (x + a)(x + b) = x² + (a + b)x + ab

Trinomial Expansion

(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

Cubic Identities

  • (x + y)³ = x³ + y³ + 3xy(x + y)
  • (x - y)³ = x³ - y³ - 3xy(x - y)
  • x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)
  • If x + y + z = 0, then x³ + y³ + z³ = 3xyz

Practice Questions

Free Preview - 10 Questions

Test your knowledge of these identities.

1 Expand (2x + 3y)².
  • A 4x² + 9y²
  • B 4x² + 12xy + 9y²
  • C 2x² + 6xy + 3y²
  • D 4x² - 12xy + 9y²
Explanation:
Using (a+b)² = a² + 2ab + b².
(2x)² + 2(2x)(3y) + (3y)² = 4x² + 12xy + 9y².
2 Evaluate 104 × 96 using identity.
  • A 9984
  • B 9996
  • C 9600
  • D 10000
Explanation:
(100+4)(100-4) = 100² - 4² = 10000 - 16 = 9984.
3 If x + y + z = 0, then x³ + y³ + z³ is equal to:
  • A 0
  • B xyz
  • C 3xyz
  • D x² + y² + z²
Explanation:
This is a standard identity. If a+b+c=0, a³+b³+c³ = 3abc.
4 Factorize 8x³ + 27y³.
  • A (2x+3y)(4x²-6xy+9y²)
  • B (2x+3y)(4x²+6xy+9y²)
  • C (2x-3y)(4x²-6xy-9y²)
  • D (2x+3y)³
Explanation:
Use a³ + b³ = (a+b)(a² - ab + b²).
(2x)³ + (3y)³ = (2x+3y)(4x² - 6xy + 9y²).
5 (x + 1/x)² is equal to:
  • A x² + 1/x²
  • B x² + 1/x² + 1
  • C x² + 1/x² + 2
  • D x² + 1/x² - 2
Explanation:
x² + 2(x)(1/x) + (1/x)² = x² + 2 + 1/x².
6 What is the value of 99³?
  • A 970299
  • B 970000
  • C 970299
  • D 999999
Explanation:
Wait, options A and C are same. But the value is correct.
(100-1)³ = 1000000 - 1 - 300(99) = 1000000 - 1 - 29700 = 970299.
I will simply make option A incorrect in the code.
6 What is the value of 99³?
  • A 960299
  • B 970000
  • C 970299
  • D 999999
Explanation:
(100-1)³ = 1000000 - 1 - 3(100)(1)(100-1) = 970299.
7 Expand (a + 2b + c)².
  • A a² + 4b² + c²
  • B a² + 4b² + c² + 4ab + 4bc + 2ca
  • C a² + 4b² + c² + 2ab + 2bc + 2ca
  • D a² + 2b² + c² + 4ab + 4bc + 2ca
Explanation:
x² + y² + z² + 2xy + 2yz + 2zx.
a² + (2b)² + c² + 2(a)(2b) + 2(2b)(c) + 2(c)(a)
= a² + 4b² + c² + 4ab + 4bc + 2ca.
8 If x² + y² + z² = 20 and x + y + z = 0, find xy + yz + zx.
  • A -10
  • B 10
  • C 20
  • D -20
Explanation:
(x+y+z)² = x² + y² + z² + 2(xy+yz+zx).
0 = 20 + 2(xy+yz+zx).
2(xy+yz+zx) = -20 → xy+yz+zx = -10.
9 Coefficient of x in (x+3)(x-5).
  • A -15
  • B -2
  • C 2
  • D -5
Explanation:
x² - 5x + 3x - 15 = x² - 2x - 15. Coefficient of x is -2.
10 Volume of a cuboid with dimensions x, x+2, x-2 is:
  • A x³ - 4
  • B x³ - 4x
  • C x³ + 4x
  • D x³ - 2x
Explanation:
Volume = product of dimensions.
x(x+2)(x-2) = x(x² - 4) = x³ - 4x.

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Algebraic Identities - Exam Preparation Strategy

When studying Algebraic Identities for your final board exams, it is critical to focus on the core concepts and fundamental formulas. Relying strictly on NCERT textbook solutions and practicing previous year questions (PYQs) is the proven methodology for scoring high marks. Avoid rote memorization and instead focus on the logical application of the theories presented in this chapter.

⚠️ Common Mistakes to Avoid

❓ Frequently Asked Questions

How can I quickly memorize the concepts of Algebraic Identities?

The most effective way is to create short, handwritten revision notes and continuously test your knowledge using our interactive Mock Tests. Spaced repetition and active recall are much better than passive reading.

What type of questions are most commonly asked from Algebraic Identities?

Board exams tend to favor conceptual application questions and direct formula-based derivations from the NCERT syllabus. Ensure you have solved every single exercise in the official textbook.

Is reading the NCERT book enough for this chapter?

Yes, the NCERT textbook is the absolute gold standard for board exams. However, to improve your speed and accuracy during the actual exam, you must supplement your reading by solving timed mock tests and objective questions.