Basic Engineering Mathematics, Fifth Edition

64 Basic Engineering Mathematics

(i) (iv) (vii)

x^2 −xy+y^2

x+y

)

x^3 + 0 + 0 +y^3

x^3 +x^2 y

−x^2 y +y^3

−x^2 y−xy^2

xy^2 +y^3

xy^2 +y^3

(i) xintox^3 goesx^2 .Putx^2 abovex^3.

(ii) x^2 (x+y)=x^3 +x^2 y

(iii) Subtract.

(iv) x into−x^2 ygoes−xy.Put−xyabove the

dividend.

(v) −xy(x+y)=−x^2 y−xy^2

(vi) Subtract.

(vii) xintoxy^2 goesy^2 .Puty^2 above the dividend.

(viii) y^2 (x+y)=xy^2 +y^3

(ix) Subtract.

Thus,

x^3 +y^3

x+y

=x^2 −xy+y^2.

Thezerosshowninthedividendarenotnormallyshown,

but are included to clarify the subtraction process and

to keep similar terms in their respective columns.

Problem 16. Divide 4a^3 − 6 a^2 b+ 5 b^3 by 2a−b

2 a^2 − 2 ab−b^2

2 a−b

)

4 a^3 − 6 a^2 b + 5 b^3

4 a^3 − 2 a^2 b

− 4 a^2 b + 5 b^3

− 4 a^2 b+ 2 ab^2

− 2 ab^2 + 5 b^3

− 2 ab^2 + b^3

4 b^3

Thus,

4 a^3 − 6 a^2 b+ 5 b^3

2 a−b

= 2 a^2 − 2 ab−b^2 , remain-

der 4b^3.

Alternatively, the answer may be expressed as

4 a^3 − 6 a^2 b+ 5 b^3

2 a−b

= 2 a^2 − 2 ab−b^2 +

4 b^3

2 a−b

Now try the following Practice Exercise

PracticeExercise 36 Basic operations in

algebra (answers on page 343)

1. Simplifypq×pq^2 r.


2. Simplify− 4 a×− 2 a.


3. Simplify 3×− 2 q×−q.


4. Evaluate 3pq− 5 qr−pqr when p= 3 ,
   q=−2andr=4.


5. Determine the value of 3x^2 yz^3 , given that
   x= 2 ,y= 1

1

2

andz=

2

3

6. Ifx=5andy=6, evaluate

23 (x−y)

y+xy+ 2 x

7. Ifa= 4 ,b= 3 ,c=5andd=6, evaluate
   3 a+ 2 b
   3 c− 2 d


8. Simplify 2x÷ 14 xy.


9. Simplify

25 x^2 yz^3

5 xyz

10. Multiply 3a−bbya+b.


11. Multiply 2a− 5 b+cby 3a+b.


12. Simplify 3a÷ 9 ab.


13. Simplify 4a^2 b÷ 2 a.


14. Divide 6x^2 yby 2xy.


15. Divide 2x^2 +xy−y^2 byx+y.


16. Divide 3p^2 −pq− 2 q^2 byp−q.


17. Simplify(a+b)^2 +(a−b)^2.

9.3 Laws of indices

The laws of indices with numbers were covered in

Chapter 7; the laws of indices in algebraic terms are

as follows:

(1) am×an=am+n

For example,a^3 ×a^4 =a^3 +^4 =a^7