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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