Basic Engineering Mathematics, Fifth Edition

Further algebra 71

(x+ 3 )is now a common factor. Thus,

x^2 (x+ 3 )− 1 (x− 3 )=(x+ 3 )(x^2 − 1 )

Now try the following Practice Exercise

PracticeExercise 40 Factorization (answers

on page 344)

Factorize and simplify the following.

1. 2x+42.2xy− 8 xz


2. pb+ 2 pc 4. 2x+ 4 xy


3. 4d^2 − 12 df^5 6. 4x+ 8 x^2


4. 2q^2 + 8 qn 8. rs+rp+rt


5. x+ 3 x^2 + 5 x^3 10. abc+b^3 c


6. 3x^2 y^4 − 15 xy^2 + 18 xy


7. 4p^3 q^2 − 10 pq^3 13. 21a^2 b^2 − 28 ab


8. 2xy^2 + 6 x^2 y+ 8 x^3 y


9. 2x^2 y− 4 xy^3 + 8 x^3 y^4


10. 28y+ 7 y^2 + 14 xy

17.

3 x^2 + 6 x− 3 xy

xy+ 2 y−y^2

18.

abc+ 2 ab

2 c+ 4

−

abc

2 c

19.

5 rs+ 15 r^3 t+ 20 r

6 r^2 t^2 + 8 t+ 2 ts

−

r

2 t

20. ay+by+a+b 21. px+qx+py+qy


21. ax−ay+bx−by


22. 2ax+ 3 ay− 4 bx− 6 by

10.4 Laws of precedence

Sometimes addition, subtraction, multiplication, divi-
sion, powers and brackets can all be involved in an
algebraic expression. With mathematics there is a defi-
nite order of precedence (first met in Chapter 1) which
we need to adhere to.
With thelaws of precedencethe order is

Brackets

Order (or pOwer)

Division

Multiplication

Addition

Subtraction

The first letter of each word spellsBODMAS.

Here are some examples to help understanding of

BODMAS with algebra.

Problem 16. Simplify 2x+ 3 x× 4 x−x

2 x+ 3 x× 4 x−x= 2 x+ 12 x^2 −x (M)

= 2 x−x+ 12 x^2

=x+ 12 x^2 (S)

orx( 1 + 12 x) by factorizing

Problem 17. Simplify(y+ 4 y)× 3 y− 5 y

(y+ 4 y)× 3 y− 5 y= 5 y× 3 y− 5 y (B)

= 15 y^2 − 5 y (M)

or 5 y( 3 y− 1 ) by factorizing

Problem 18. Simplifyp+ 2 p×( 4 p− 7 p)

p+ 2 p×( 4 p− 7 p)=p+ 2 p×− 3 p (B)

=p− 6 p^2 (M)

orp( 1 − 6 p) by factorizing

Problem 19. Simplifyt÷ 2 t+ 3 t− 5 t

t÷ 2 t+ 3 t− 5 t=

t

2 t

+ 3 t− 5 t (D)

=

1

2

+ 3 t− 5 t by cancelling

=

1

2

− 2 t (S)

Problem 20. Simplifyx÷( 4 x+x)− 3 x

x÷( 4 x+x)− 3 x=x÷ 5 x− 3 x (B)

=

x

5 x

− 3 x (D)

=

1

5

− 3 x by cancelling

Problem 21. Simplify 2y÷( 6 y+ 3 y− 5 y)

2 y÷( 6 y+ 3 y− 5 y)= 2 y÷ 4 y (B)

=

2 y

4 y

(D)

=

1

2

by cancelling