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Factorize each of the following expressions:(i) (ax + by)2 + (bx – ay)2(ii) 16 (a – b)3 – 24 (a – b)2(iii) ab (x2 + 1) + x (a2 + b2)(iv) a2x2 + (ax2 + 1)x + a(v) a (a – 2b – c) + 2bc |
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Answer» (i) (ax + by)2 + (bx – ay)2 = a2x2 + b2y2 + 2axby + b2x2 + a2y2 – 2axby = a2x2 + b2y2 + b2x2 + a2y2 = a2x2 + a2y2 + b2y2 + b2x2 = a2 (x2 + y2) + b2 (x2 + y2) = (a2 + b2) (x2 + y2) (ii) 16 (a – b)3 – 24 (a – b)2 = 8 (a – b)2 [2 (a – b) – 3] = 8 (a – b)2 (2a – 2b – 3) (iii) ab (x2 + 1) + x (a2 + b2) = abx2 + ab + xa2 + xb2 = abx2 + xa2 + ab + xb2 = ax (bx + a) + b (bx + a) = (ax + b) (bx + a) (iv) a2x2 + (ax2 + 1)x + a = a2x2 + ax3 + x + a = ax2 (a + x) + 1 (x + a) = (x + a) (ax2 + 1) (v) a (a – 2b – c) + 2bc = a2 – 2ab – ac + 2bc = a (a – 2b) – c (a – 2b) = (a – 2b) (a – c) |
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