(a + b) ²= a ² + 2ab + b ²

• (a + b) ²= (a – b)  ² + 4ab

 • (a – b)  ² = a ²  – 2ab + b ²  • (a – b)  ² = (a + b)  ² – 4ab

 • a ²  + b ²  = (a + b)  ² – 2ab

• a2 + b2 = (a – b)  ² + 2ab

 • a ²  – b ²  = (a + b) – (a – b)

 • 2 (a ² +b ²) = (a + b)  ²   + (a – b)  ²

  • (a + b + c)  ² = (a ²  + b ²  + c ²) + 2 (ab + bc + ca)

• (a ²  + b ²  + c ²) = (a + b + c)  ² – 2(ab + bc + ca)

• 2 (ab + bc + ca) = (a + b + c)  ² – (a² + b² + c²)

• (a + b)³= a³ + 3a²b + 3ab² + b³

• (a + b) ³= a³ + b³ + 3ab (a + b)

• (a – b) ³= a³ – 3a²b + 3ab² – b3

• (a – b) ³= a³ – b³ – 3ab (a – b)

• a³ + b³ = (a + b) (a² – ab + b²)

• a³ + b³ = (a + b) ³– 3ab (a + b)

• a³ – b³ = (a – b) (a² + ab + b²)

• a³ – b³ = (a – b) ³+ 3ab (a – b)

 • (a + b + c) ³= a³ + b³ + c³ + 3 (a + b) (b + c) (c + a)

• a³ + b³+ c³ – 3abc = (a + b + c) (a² + b² + c²– ab – bc – ca)

 • a3 + b3 + c3 – 3abc =  (a + b + c) { (a – b)² + (b – c) ²+ (c – a) ²}

• 4ab = (a + b) ²– (a – b) ²  

• (a + b) ²= a ² + 2ab + b ²

• (a + b) ²= (a – b)  ² + 4ab

 • (a – b)  ² = a ²  – 2ab + b ²  • (a – b)  ² = (a + b)  ² – 4ab

 • a ²  + b ²  = (a + b)  ² – 2ab

• a2 + b2 = (a – b)  ² + 2ab

 • a ²  – b ²  = (a + b) – (a – b)

 • 2 (a ² +b ²) = (a + b)  ²   + (a – b)  ²

  • (a + b + c)  ² = (a ²  + b ²  + c ²) + 2 (ab + bc + ca)

• (a ²  + b ²  + c ²) = (a + b + c)  ² – 2(ab + bc + ca)

• 2 (ab + bc + ca) = (a + b + c)  ² – (a² + b² + c²)

• (a + b)³= a³ + 3a²b + 3ab² + b³

• (a + b) ³= a³ + b³ + 3ab (a + b)

• (a – b) ³= a³ – 3a²b + 3ab² – b3

• (a – b) ³= a³ – b³ – 3ab (a – b)

• a³ + b³ = (a + b) (a² – ab + b²)

• a³ + b³ = (a + b) ³– 3ab (a + b)

• a³ – b³ = (a – b) (a² + ab + b²)

• a³ – b³ = (a – b) ³+ 3ab (a – b)

 • (a + b + c) ³= a³ + b³ + c³ + 3 (a + b) (b + c) (c + a)

• a³ + b³+ c³ – 3abc = (a + b + c) (a² + b² + c²– ab – bc – ca)

 • a3 + b3 + c3 – 3abc =  (a + b + c) { (a – b)² + (b – c) ²+ (c – a) ²}

• 4ab = (a + b) ²– (a – b) ²  

• (x + a) (x + b) = x² + (a + b) x + ab

• (x + a) (x – b) = x² + (a – b) x – ab

• (x – a) (x + b) = x² + (b – a) x – ab

• (x – a) (x – b) = x² – (a + b) x + ab

 • (x + p) (x + q) (x + r) = x³ + (p + q + r) x² + (pq + qr + rp) x +pqr

• bc (b – c) + ca (c – a) + ab (a – b) = – (b – c) (c – a) (a – b)

• a²(b – c) + b² (c – a) + c² (a – b) = – (b – c) (c – a) (a – b)

 • a(b² – c²) + b (c² – a²) + c (a² – b²) = (b – c) (c – a) (a – b)

•  a³ (b – c) + b³ (c – a) + c³ (a – b) = – (b – c) (c – a) (a – b) (a + b + c)

• b²c²(b²–c²) +c² a² (c²–a²) + a²b² (a²–b²) = – (b–c) (c–a) (a–b) (b+c) (c+a) (a+b)

• (ab + bc + ca) (a + b + c) – abc = (a + b) (b + c) (c + a)

• (b + c) (c + a) (a + b) + abc = (a + b +c) (ab + bc + ca)

উদাহরণ-১ঃ

যদি a³ – b³ = 513 এবং a – b = 3 হয়, তবে ab এর মান কত?

সমাধানঃ

আমরা জানি, (a – b)³ = a3 – b3 – 3ab(a – b)

=> 3³ = 513 – 3ab*3

=> 9ab=513-27

=> ab=54 (Ans.)

উদাহরণ-২ঃ

x + y = 2, x² + y² = 4 হলে x³ + y³ = কত?

সমাধানঃ

আপনি বর্গের একটি সূত্র জানেন,

 (x+ y) ²= x ² + 2xy + y ²

মান বসানঃ

2²=4+ 2xy

xy =0

এখন

x³ + y³ এর মান বের করতে হবে।

আমরা কিউবের সূত্র জানিঃ

(x+y)³= x³ + y³+ 3xy(x+y)

মান বসানঃ

(2)³ = x³ + y³+ 3x0x2

8 = x³ + y³

উদাহরণ-৩ঃ

a+1/a=3 হলa³+1/a³ এর মান কত?

সমাধানঃ

a³+1/a³

=(a+1/a)³-3 a. 1/a(a+1/a)

=3³-3*3

=27-9

=18

উদাহরণ-৪ঃ

x-y=2 এবং xy=3 হলে x+y এর মান?

সমাধানঃ

(x+y)²=(x-y)²+4xy

(x+y)²=2²+4×3

(x+y)²=16

(x+y)=±4

উদাহরণ-৫ঃ

у – এর মান কত হলে 16x²-xy+25 একটি পূৰ্ণবর্গ রাশি হবে?

সমাধানঃ

16x² — ху+25

 y=40

(4x)² –40x+5²

=(4x)²–2.4x.5+5²

==(4x)² – 40x+5²

=(4x–5)²

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