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Addition and Subtraction of Algebraic Expressions

Simplify:1 x+...

Question

Simplify:
(1) (x + y)³− (x − y)³
(2) (3a + 5b)³− (3a − 5b)³
(3) (a + b)³− a³− b³
(4) p³− (p + 1)³
(5) (3xy − 2ab)³ − (3xy + 2ab)³

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Solution

It is known that,
$a^{3} - b^{3} = (a - b) (a^{2} + b^{2} + a b)$
$(1) {(x + y)}^{3} - {(x - y)}^{3} = \{(x + y) - (x - y)\} \{{(x + y)}^{2} + {(x - y)}^{2} + (x + y) (x - y)\} = 2 y (x^{2} + y^{2} + 2 x y + x^{2} + y^{2} - 2 x y + x^{2} - y^{2}) = 2 y (3 x^{2} + y^{2})$
$(2) {(3 a + 5 b)}^{3} - {(3 a - 5 b)}^{3} = \{(3 a + 5 b) - (3 a - 5 b)\} \{{(3 a + 5 b)}^{2} + {(3 a - 5 b)}^{2} + (3 a + 5 b) (3 a - 5 b)\} = 10 b (9 a^{2} + 25 b^{2} + 30 a b + 9 a^{2} + 25 b^{2} - 30 a b + 9 a^{2} - 25 b^{2}) = 10 b (27 a^{2} + 25 b^{2})$
$(3) {(a + b)}^{3} - a^{3} - b^{3} = {(a + b)}^{3} - (a^{3} + b^{3}) = {(a + b)}^{3} - \{(a + b) (a^{2} + b^{2} - a b)\} = (a + b) \{{(a + b)}^{2} - (a^{2} + b^{2} - a b)\} = (a + b) (a^{2} + b^{2} + 2 a b - a^{2} - b^{2} + a b) = (a + b) (3 a b) = 3 a^{2} b + 3 a b^{2}$
$(4) p^{3} - {(p + 1)}^{3} = \{p - (p + 1)\} \{{(p)}^{2} + {(p + 1)}^{2} + (p) \times (p + 1)\} = (- 1) (p^{2} + p^{2} + 1 + 2 p + p^{2} + p) = (- 1) (3 p^{2} + 3 p + 1) = - 3 p^{2} - 3 p - 1$
$(5) {(3 x y - 2 a b)}^{3} - {(3 x y + 2 a b)}^{3} = \{(3 x y - 2 a b) - (3 x y + 2 a b)\} \{{(3 x y - 2 a b)}^{2} + {(3 x y + 2 a b)}^{2} + (3 x y - 2 a b) (3 x y + 2 a b)\} = (- 4 a b) (9 x^{2} y^{2} + 4 a^{2} b^{2} - 12 a b x y + 9 x^{2} y^{2} + 4 a^{2} b^{2} + 12 a b x y + 9 x^{2} y^{2} - 4 a^{2} b^{2}) = (- 4 a b) (27 x^{2} y^{2} + 4 a^{2} b^{2}) = - 108 x^{2} y^{2} a b - 16 a^{3} b^{3}$

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