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Chapter 08 – Factorisation | Class 9 Flash Formula
Square of a Sum
(a + b)² = a² + 2ab + b²
(a + b)² = (a − b)² + 4ab
Example:
(x + 2)² = x² + 2(x)(2) + 2²
= x² + 4x + 4
Square of a Difference
(a − b)² = a² − 2ab + b²
(a − b)² = (a + b)² − 4ab
Example:
(x - 2)² = x² - 2(x)(2) + 2²
= x² - 4x + 4
Sum of Squares of Sum and Difference
(a + b)² + (a − b)² = 2(a² + b²)
Example :
(x + 3)² + (x − 3)²
= (x² + 6x + 9) + (x² − 6x + 9)
= 2(x² + 9)
= 2x² + 18
Difference of Squares of Sum and Difference
(a + b)² − (a − b)² = 4ab
Explanation
Expand both squares:
(a+b)² = a² + 2ab + b²
(a-b)² = a² - 2ab + b²
Substitute:
(a + b)² − (a − b)² = a² + 2ab + b² - (a² - 2ab + b²)
= a² + 2ab + b² - a² + 2ab - b²
= 4ab
Sum of Squares of Reciprocal Expressions
(a + 1\over \text{a²})² + (a − 1\over \text{a²})² = 2(a² + 1\over \text{a²})
Explanation:
(a + 1/a)² + (a − 1/a)²
Expand both squares:
(a + 1/a)² = a² + 2 + 1/a²
(a − 1/a)² = a² − 2 + 1/a²
Substitute:
(a + 1/a)² + (a − 1/a)² = (a² + 2 + 1/a²) + (a² − 2 + 1/a²)
= 2a² + 2/a²
= 2(a² + 1/a²)
Difference of Squares of Reciprocal Expressions
(a + 1\over \text{a²})² − (a − 1\over \text{a²})² = 4
Explanation:
(a + 1/a)² − (a − 1/a)²
= (a + 1/a)² − (a − 1/a)²
= [a² + 2(a)(1/a) + (1/a)²] − [a² − 2(a)(1/a) + (1/a)²]
= [a² + 2 + 1/a²] − [a² − 2 + 1/a²]
= a² + 2 + 1/a² − a² + 2 − 1/a² = 4
Difference of Two Squares
a2 − b2 = (a + b)(a − b)
Example:
x² − 25
= x² − 5²
= (x + 5)(x − 5)
Square of Three-Term Expression
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
Example :
(x + y − 5)²
= x² + y² + (−5)² + 2(xy + y(−5) + (−5)x)
= x² + y² + 25 + 2(xy − 5y − 5x)
= x² + y² + 25 + 2xy − 10y − 10x
= x² + y² + 2xy − 10x − 10y + 25
Cube of a Sum
(i) (a + b)³ = a³ + b³ + 3ab(a + b)
(ii) (a + b)³ = a³ + 3a²b + 3ab² + b³
Example :
(x + 4)³
= x³ + 3x²(4) + 3x(4²) + 4³
= x³ + 12x² + 48x + 64
Cube of a Difference
(i) (a − b)³ = a³ − b³ − 3ab(a − b)
(ii) (a − b)³ = a³ − 3a²b + 3ab² − b³
Example (Using Formula i)
(x − 7)³
= x³ − 3x²(7) + 3x(7²) − 7³
= x³ − 21x² + 147x − 343
Difference of Two Cubes
(i) a³ − b³ = (a − b)(a² + ab + b²)
(ii) a³ − b³ = (a − b)³ + 3ab(a − b)
Example (Using Formula i)
x³ − 125
= x³ − 5³
= (x − 5)(x² + 5x + 25)
Example (Using Formula ii)
x³ − 125
= x³ − 5³
= (x − 5)³ + 3(x)(5)(x − 5)
= (x − 5)³ + 15x(x − 5)
Sum of Two Cubes
(i) a³ + b³ = (a + b)(a² − ab + b²)
(ii) a³ + b³ = (a + b)³ − 3ab(a + b)
Example:
x³ + 343
= x³ + 7³
= (x + 7)(x² − 7x + 49)
Sum of Three Cubes Identity
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)
Also, If a + b + c = 0, a³ + b³ + c³ = 3abc
Example:
x³ + 8a³ + 27 − 18ax
= x³ + (2a)³ + 3³ − 3(x)(2a)(3)
= (x + 2a + 3)[x² + (2a)² + 3² − x(2a) − (2a)(3) − 3x]
= (x + 2a + 3)(x² + 4a² + 9 − 2ax − 6a − 3x)
Product of Two Binomials
(i) (x + a)(x + b) = x² + (a + b)x + ab
(ii) (x + a)(x − b) = x² + (a − b)x − ab
(iii) (x − a)(x + b) = x² − (a − b)x − ab
(iv) (x − a)(x − b) = x² − (a + b)x + ab
Example
(x + 3)(x + 5)
= x² + (3 + 5)x + (3)(5)
= x² + 8x + 15
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