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Mathematics Question on What is Factorisation?

Factorise the following expressions

  1. p2+6p+8p^ 2 + 6p + 8
  2. q2−10q+21q^ 2 - 10q + 21
  3. p2+6p−16p^ 2 + 6p - 16
Answer

(i) p2+6p+8p^2+ 6p + 8
It can be observed that, 8=4×28 = 4 \times 2 and 4+2=64 + 2 = 6
∴ p2+6p+8=p2+2p+4p+8p^2+ 6p + 8 = p^2+ 2p + 4p + 8
= p(p+2)+4(p+2)p(p + 2) + 4(p + 2)
= (p+2)(p+4)(p + 2) (p + 4)

(ii) q2−10q+21q^2- 10q + 21
It can be observed that, 21=(−7)×(−3)21 = (-7) \times (-3) and (−7)+(−3)=−10(-7) + (-3) = - 10
∴ q2−10q+21=q2−7q−3q+21q^2- 10q + 21 = q^2- 7q - 3q + 21
= q(q−7)−3(q−7)q(q - 7) - 3(q - 7)
= (q−7)(q−3)(q - 7) (q - 3)

(iii) p2+6p−16p^2+ 6p - 16
It can be observed that, 16=(−2)×816 = (-2) \times 8 and 8+(−2)=68 + (-2) = 6
p2+6p−16=p2+8p−2p−16p^2+ 6p - 16 = p^2+ 8p - 2p - 16
= p(p+8)−2(p+8)p(p + 8) - 2(p + 8)
= (p+8)(p−2)(p + 8) (p - 2)