Question

Simplify.
(i) (x²– 5) (x + 5) + 25
(ii) (a²+ 5) (b³+ 3) + 5
(iii) (t + s²) (t² – s)
(iv) (a + b) (c – d) + (a – b) (c + d) + 2 (ac + bd)
(v) (x + y) (2x + y) + (x + 2y) (x – y)
(vi) (x + y) (x²– xy + y²)
(vii) (1.5x – 4y) (1.5x + 4y + 3) – 4.5x + 12y
(viii) (a + b + c) (a + b – c)

Answer 1

(i) (x²– 5) (x + 5) + 25
= x³ + 5x² – 5x – 25 + 25
= x³ + 5x² – 5x

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(ii) (a²+ 5) (b³+ 3) + 5
= a²b³ + 3a² + 5b³ + 15 + 5
= a²b³ + 5b³ + 3a² + 20

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(iii) (t + s²) (t² – s)
= t (t² – s) + s²(t² – s)
= t³ – st + s²t² – s³
= t³ – s³ – st + s²t²

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(iv) (a + b) (c – d) + (a – b) (c + d) + 2 (ac + bd)
= (a + b) (c – d) + (a – b) (c + d) + 2 (ac + bd)
= (ac – ad + bc – bd) + (ac + ad – bc – bd) + (2ac + 2bd)
= ac – ad + bc – bd + ac + ad – bc – bd + 2ac + 2bd
= 4ac

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(v) (x + y) (2x + y) + (x + 2y) (x – y)
= 2x² + xy + 2xy + y² + x² – xy + 2xy – 2y²
= 3x² + 4xy – y²

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(vi) (x + y) (x²– xy + y²)
= x³ – x²y + xy² + x²y – xy² + y³
= x³ + y³

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(vii) (1.5x – 4y) (1.5x + 4y + 3) – 4.5x + 12y
= 2.25x² + 6xy + 4.5x – 6xy – 16y² – 12y – 4.5x + 12y 
= 2.25x² – 16y²

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(viii) (a + b + c) (a + b – c)
= a² + ab – ac + ab + b² – bc + ac + bc – c²
= a² + b² – c² + 2ab