Question

simplify (x+y)(x²-xy+y²)
(x-y)(x²+xy+y²
½(x+y+2)[(x-y)²+(y-z)²+z-x)²)]​

Answer 1

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Answer 2

${\large{\underline{\underline{\pmb{\frak{Answer \: 1 :-}}}}}}$

$\huge\bold\pink{(x + y)(x² - xy + y²)}$

$\dashrightarrow$x (x² - xy + y²) - y (x² - xy + y²)

$\dashrightarrow$x³ - x²y + xy² + x²y - xy² + y³

• - x²y + x²y and - xy² + xy² get cancelled

$\longrightarrow$ x³ + y³

${\large{\underline{\underline{\pmb{\frak{Answer \: 2 :-}}}}}}$

$\huge\bold\red{(x - y)(x² + xy + y²)}$

$\dashrightarrow$x (x² + xy + y²) - y (x² + xy + y²)

$\dashrightarrow$x³ + x²y + xy² - x²y - xy² - y³

• + x²y - x²y and + xy² - xy² get cancelled

$\longrightarrow$ x³ - y³

${\large{\underline{\underline{\pmb{\frak{Answer \: 3 :-}}}}}}$

$\bold\orange{x³ + y³ + z³ - 3xyz = ½ (x + y + z)[(x - y)² + (y - z)² + (z - x)²]}$

$\dashrightarrow$RHS = $\frac{1}{2}$(x + y + z)[(x - y)² + (y - z)² + (z - x)²]

$\dashrightarrow$$\frac{1}{2}$(x + y + z) (x² + y² - 2xy + y² + z² - 2xy + z² + x² - 2zx)

• Take 2 common

$\dashrightarrow$$\frac{2}{2}$ (x + y + z) (x² + y² + z² - xy - yz - zx)

$\dashrightarrow$x (x² + y² + z² - xy - yz - zx) + y (x² + y² + z² - xy - yz - zx) + z (x² + y² + z² - xy - yz - zx)

$\dashrightarrow$x³ + xy² + xz² - x²y - xyz - zx² + x²y + y³ + z²y - xy² - y²z - xyz + x²z - zy² + z³ - xyz - yz² - z²x

• Many terms get cancelled
• And we left with -
• x³ + y³ + z³ - xyz - xyz - xyz

$\longrightarrow$ x³ + y³ + z³ - 3xyz