Basic Engineering Mathematics, Fifth Edition

Further algebra 69

Problem 7. Remove the brackets from the

expression and simplify 2[x^2 − 3 x(y+x)+ 4 xy]

2[x^2 − 3 x(y+x)+ 4 xy]=2[x^2 − 3 xy− 3 x^2 + 4 xy]

(Whenever more than one type of brackets is involved,

alwaysstart with the inner brackets)

=2[− 2 x^2 +xy]

=− 4 x^2 + 2 xy

= 2 xy− 4 x^2

Problem 8. Remove the brackets and simplify the

expression 2a−[3{ 2 ( 4 a−b)− 5 (a+ 2 b)}+ 4 a]

(i) Removing the innermost brackets gives

2 a−[3{ 8 a− 2 b− 5 a− 10 b}+ 4 a]

(ii) Collecting together similar terms gives

2 a−[3{ 3 a− 12 b}+ 4 a]

(iii) Removing the ‘curly’ brackets gives

2 a−[9a− 36 b+ 4 a]

(iv) Collecting together similar terms gives

2 a−[13a− 36 b]

(v) Removing the outer brackets gives

2 a− 13 a+ 36 b

(vi) i.e.− 11 a+ 36 b or 36 b− 11 a

Now try the following Practice Exercise

PracticeExercise 39 Brackets(answerson

page 344)

Expand the brackets in problems 1 to 28.

1. (x+ 2 )(x+ 3 ) 2. (x+ 4 )( 2 x+ 1 )


2. ( 2 x+ 3 )^2 4. ( 2 j− 4 )(j+ 3 )


3. ( 2 x+ 6 )( 2 x+ 5 ) 6. (pq+r)(r+pq)


4. (a+b)(a+b) 8. (x+ 6 )^2


5. (a−c)^2 10. ( 5 x+ 3 )^2


6. ( 2 x− 6 )^2 12. ( 2 x− 3 )( 2 x+ 3 )


7. ( 8 x+ 4 )^2 14. (rs+t)^2


8. 3a(b− 2 a) 16. 2x(x−y)


9. ( 2 a− 5 b)(a+b)


10. 3( 3 p− 2 q)−(q− 4 p)


11. ( 3 x− 4 y)+ 3 (y−z)−(z− 4 x)


12. ( 2 a+ 5 b)( 2 a− 5 b)


13. (x− 2 y)^2 22.( 3 a−b)^2


14. 2x+[y−( 2 x+y)]


15. 3a+2[a−( 3 a− 2 )]


16. 4[a^2 − 3 a( 2 b+a)+ 7 ab]


17. 3[x^2 − 2 x(y+ 3 x)+ 3 xy( 1 +x)]


18. 2−5[a(a− 2 b)−(a−b)^2 ]


19. 24p−[2{ 3 ( 5 p−q)− 2 (p+ 2 q)}+ 3 q]

10.3 Factorization

Thefactorsof 8 are 1, 2, 4 and 8 because 8 divides by

1, 2, 4 and 8.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 because 24

divides by 1, 2, 3, 4, 6, 8, 12 and 24.

Thecommon factorsof 8 and 24 are 1, 2, 4 and 8 since

1, 2, 4 and 8 are factors of both 8 and 24.

The highest common factor (HCF) is the largest

number that divides into two or more terms.

Hence, the HCF of 8 and 24 is 8, as explained in

Chapter 1.

When twoor more terms inan algebraic expression con-

tain a common factor, then this factor can be shown

outside of a bracket. For example,

df+dg=d(f+g)

which is just the reverse of

d(f+g)=df+dg

This process is calledfactorization.

Here are some worked examples to help understanding

of factorizing in algebra.

Problem 9. Factorizeab− 5 ac

aiscommontobothtermsaband− 5 ac.ais there-

fore taken outside of the bracket. What goes inside the

bracket?