Factorise :

$(i) a^{2} - a b - 3 a + 3 b (i i) x^{2} y - x y^{2} + 5 x - 5 y (i i i) a^{2} - a b (1 - b) - b^{3} (i v) x y^{2} + (x - 1) y - 1 (v) (a x + b y)^{2} + (b x - a y)^{2} (v i) a b (x^{2} + y^{2}) - x y (a^{2} + b^{2}) (v i i) m - 1 - (m - 1)^{2} + a m - a$

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Solution

$(i) a^{2} - a b - 3 a + 3 b = a (a - b) - 3 (a - b) = (a - b) (a - 3) (i i) x^{2} y - x y^{2} + 5 x - 5 y = x y (x - y) + 5 (x - y) = (x - y) (x y + 5) (i i i) a^{2} - a b (1 - b) - b^{3} = a^{2} - a b + a b^{2} - b^{3} = a (a - b) + b^{2} (a - b) = (a - b) (a + b)^{2} (i v) x y^{2} + (x - 1) y - 1 = x y^{2} + x y - y - 1 = x y (y + 1) - 1 (y + 1) = (x y - 1) (y + 1) (v) (a x + b y)^{2} + (b x - a y)^{2} = a^{2} x^{2} + b^{2} y^{2} + 2 a b x y + b^{2} x^{2} + a^{2} y^{2} - 2 a b x y = a^{2} x^{2} + b^{2} y^{2} + b^{2} x^{2} + . y^{2} - a^{2} x^{2} + a^{2} y^{2} + b^{2} x^{2} + b^{2} y^{2} - a^{2} (x^{2} + y^{2}) + b^{2} (x^{2} + y^{2}) = (x^{2} + y^{2}) (a^{2} + b^{2}) (v i) a b (x^{2} + y^{2}) - x y (a^{2} + b^{2}) = a b x^{2} + a b y^{2} - a^{2} x y - b^{2} x y = a b x^{2} - a^{2} x y + a b y^{2} - b^{2} x y = a b x^{2} - a^{2} x y - b^{2} x y + a b y^{2} = a x (b x - a y) - b y (b x - a y) = (b x - a y) (a x - b y) (v i i) m - 1 - (m - 1)^{2} + a m - a = (m - 1) - (m - 1)^{2} + a (m - 1) = (m - 1) (1 - (m - 1) + a) = (m - 1) (1 - m + 1 + a) = (m - 1) (2 - m + a)$

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