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

=x² (x + 5) − 5 (x + 5) + 25

=x³ + 5x² − 5x −25 + 25

=x³ + 5x² − 5x

(ii)(a²+ 5) (b³ + 3) + 5

=a² (b³ + 3) + 5 (b³+ 3) + 5

=a²b³ + 3a² +5b³ + 15 + 5

=a²b³ + 3a² + 5b³+ 20

(iii)(t + s²)(t² − s)

= t(t² − s) + s² (t²− s)

=t³ − st + s²t²− s³

(iv)(a + b)(c − d) + (a − b) (c +d) + 2 (ac + bd)

= a(c − d) + b (c − d) +a (c + d) − b (c + d)+ 2 (ac + bd)

=ac − ad + bc − bd + ac+ ad − bc − bd + 2ac + 2bd

=(ac + ac + 2ac) + (ad − ad)+ (bc − bc) + (2bd − bd −bd)

=4ac

(v)(x + y)(2x + y) + (x + 2y) (x − y)

= x(2x + y) + y (2x + y) + x(x − y) + 2y (x − y)

=2x² + xy + 2xy + y²+ x² − xy + 2xy − 2y²

=(2x² + x²) + (y²− 2y²) + (xy + 2xy − xy+ 2xy)

=3x² − y² + 4xy

(vi)(x + y)(x² − xy + y²)

= x(x² − xy + y²) + y(x² − xy + y²)

=x³ − x²y + xy²+ x²y − xy² + y³

=x³ + y³ + (xy²− xy²) + (x²y −x²y)

=x³ + y³

(vii)(1.5x −4y) (1.5x + 4y + 3) − 4.5x + 12y

=1.5x (1.5x + 4y + 3) − 4y (1.5x+ 4y + 3) − 4.5x + 12y

=2.25 x² + 6xy + 4.5x − 6xy− 16y² − 12y − 4.5x+ 12y

=2.25 x² + (6xy − 6xy) + (4.5x− 4.5x) − 16y² + (12y −12y)

=2.25x² − 16y²

(viii)(a + b+ c) (a + b − c)

= a(a + b − c) + b (a + b− c) + c (a + b − c)

=a² + ab − ac + ab + b²− bc + ca + bc − c²

=a² + b² − c²+ (ab + ab) + (bc − bc) + (ca− ca)

=a² + b² − c²+ 2ab