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Mathematics Question on What is Factorisation?

Factorise

  1. x2+xy+8x+8yx^ 2 + xy + 8x + 8y
  2. 15xy−6x+5y−215xy - 6x + 5y - 2
  3. ax+bx−ay−byax + bx - ay - by
  4. 15pq+15+9q+25p15pq + 15 + 9q + 25p
  5. z−7+7xy−xyzz - 7 + 7xy - xyz
Answer

(i) x2+xy+8x+8y=x×x+x×y+8×x+8×yx^2 + xy + 8x + 8y = x \times x + x \times y + 8 \times x + 8 \times y
= x(x+y)+8(x+y)x (x + y) + 8 (x + y)
= (x+y)(x+8)(x + y) (x + 8)

(ii) 15xy−6x+5y−2=3×5×x×y−3×2×x+5×y−215xy - 6x + 5y - 2 = 3 \times 5 \times x \times y - 3 \times 2 \times x + 5 \times y - 2
= 3x  (5y−2)+1(5y−2)3x \;(5y - 2) + 1 (5y - 2)
= (5y−2)(3x+1)(5y - 2) (3x + 1)

(iii) ax+bx−ay−by=a×x+b×x−a×y−b×yax + bx - ay - by = a \times x + b \times x - a \times y - b \times y
= x(a+b)−y(a+b)x (a + b) - y (a + b)
= (a+b)(x−y)(a + b) (x - y)

(iv) 15pq+15+9q+25p=15pq+9q+25p+1515pq + 15 + 9q + 25p = 15pq + 9q + 25p + 15
= 3×5×p×q+3×3×q+5×5×p+3×53 \times 5 \times p \times q + 3 \times 3 \times q + 5 \times 5 \times p + 3 \times 5
= 3q(5p+3)+5(5p+3)3q (5p + 3) + 5 (5p + 3)
= (5p+3)(3q+5)(5p + 3) (3q + 5)

(v) z−7+7xy−xyz=z−x×y×z−7+7×x×yz - 7 + 7xy - xyz = z - x \times y \times z - 7 + 7 \times x \times y
= z(1−xy)−7(1−xy)z (1 - xy) - 7 (1 - xy)
= (1−xy)(z−7)(1 - xy) (z - 7)