Question

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

Answer 1

(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)