Question

Factorise the polynomial (a+b)³+(b+c)³+(c+a)³-3(a+b)(b+c)(c+a)​

Answer 1

$\large\text{\fcolorbox{aqua}{gold}{\red{Given:-}}}$

Factorise the polynomial (a+b)³+(b+c)³+(c+a)³-3(a+b)(b+c)(c+a)

$\large\text{\fcolorbox{aqua}{yellow}{\red{Solution:-}}}$

(a+b)³+(b+c)³+(c+a)³-3(a+b)(b+c)(c+a)

Let (a+b) = A, (b+c) = B, (c+a) = C

→ A³+ B³ + C³- 3ABC

Using identity,

x³+ y³ +z³-3xyz = (x+y+z)(x²+y²+z²- xy- yz- za)

→ (A+B+C) (A²+B²+C²-AB - BC - CA)

By substituting the values of A,B and C.

→ {a+b+b+c+c+a} {(a+b)² + (b+c)² + (c+a)² - (a+b)(b+c) - (b+c) (c+a) - (c+a) (a+b)}

→ (2a+2b+2c) (a²+b²+c²+ 2ab +b²+ c²+ 2bc+ c²+ a²+ 2ac- ab- ac- b²- bc- bc- ab- c²- ac- ac- bc-a²- ab)

$\green{\sf{→ 2(a+ b+ c) (a²+ b²+ c²- ab- bc- ca)}}$

$\large\text{\fcolorbox{gold}{lime}{\red{Hope it's help you}}}$