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Expansion of (x+y+z)^2

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Question

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(i) $(3 x - 4 y + 5 z)^{2}$

(ii) $(2 a - 5 b - 4 c)^{2}$

(iii) $(5 x + 3 y)^{3}$

(iv) $(6 a - 7 b)^{3}$

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Solution

(i) $(3 x - 4 y + 5 z)^{2} (3 x)^{2} + (- 4 y)^{2} + (5 z)^{2} + 2 (3 x) (- 4 y) + 2 (- 4 y) (5 z) + 2 (5 z) (3 x) = 9 x^{2} + 16 y^{2} + 25 z^{2} - 24 x y - 40 y z + 30 z x$

(ii) $(2 a - 5 b - 4 c)^{2} = (2 a)^{2} + (- 5 b)^{2} + (- 4 c)^{2} + 2 (2 a) (- 5 b) + 2 (- 5 b) (- 4 c) + 2 (- 4 c) (2 a) = 4 a^{2} + 25 b^{2} + 16 c^{2} - 20 a b + 40 b c - 16 c a$

(iii) $(5 x + 3 y)^{3} = (5 x)^{3} + (3 y)^{3} + 3 (5 x) (3 y) (5 x + 3 y) = 125 x^{2} + 27 y^{3} + 45 x y (5 x + 3 y) = 125 x^{3} + 27 y^{3} + 225 x^{2} y + 135 x y^{2}$

(iv) $(6 a - 7 b)^{3} = (6 a)^{3} - (7 b)^{3} - 3 (6 a) (7 b) (6 a - 7 b) = 216 a^{3} - 343 b^{3} - 126 a b (6 a - 7 b) = 216 a^{3} - 343 b^{3} - 756 a^{2} b + 882 a b^{2} = 216 a^{3} - 756 a^{2} b + 882 a b^{2} - 343 b^{3}$

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Expand
(i) $(2 a - 5 b - 7 c)^{2}$ (ii) $(- 3 a + 4 b - 5 c)^{2}$ (iii) $(\frac{1}{2} a - \frac{1}{4} b + 2)^{2}$

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2 a - b, 4 a - 3 b, 6 a - 5 b

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