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)