64 Basic Engineering Mathematics
(i) (iv) (vii)
x 2 − x y + y 2
x + y
x 3 + 0 + 0 + y 3
x 3 + x 2 y
−x
2 y
+ y
3
−x 2 y − x y 2
x y
2
+ y
3
x y 2 + y 3
.
.
(i) x into x 3 goes x 2 . Put x 2 above x 3 .
(ii) x 2 (x + y) = x 3 + x 2 y
(iii) Subtract.
(iv) x into −x
2 y goes −x y. Put −x y above the
dividend.
(v) −x y(x + y) = −x 2 y − x y 2
(vi) Subtract.
(vii) x into x y 2 goes y 2 . Put y 2 above the dividend.
(viii) y 2 (x + y) = x y 2 + y 3
(ix) Subtract.
Thus,
x 3 + y 3
x + y
= x 2 − xy + y 2 .
The zeros shown in the dividend are not normally shown,
but are included to clarify the subtraction process and
to keep similar terms in their respective columns.
Problem 16. Divide 4a 3 − 6a 2 b + 5b 3 by 2a − b
2a 2 − 2ab − b 2
2a − b
4a 3 − 6a 2 b
+ 5b 3
4a 3 − 2a 2 b
−4a
2 b
+ 5b
3
−4a 2 b + 2ab 2
−2ab
2
+ 5b
3
−2ab 2 + b 3
4b
3
Thus,
4a 3 − 6a 2 b + 5b
3
2a − b
= 2a 2 − 2ab − b 2 , remainder 4b
3 .
Alternatively, the answer may be expressed as
4a 3 − 6a 2 b + 5b
3
2a − b
= 2a
2
− 2ab − b
2
+
4b
3
2a − b
Now try the following Practice Exercise
Practice Exercise 36 Basic operations in
algebra (answers on page 343)
1. Simplify pq × pq 2 r.
2. Simplify −4a × −2a.
3. Simplify 3 × −2q × −q.
4. Evaluate 3 pq − 5qr − pqr when p = 3,
q = −2 and r = 4.
5. Determine the value of 3x 2 yz 3 , given that
x = 2, y = 1
1
2
and z =
2
3
6. If x = 5 and y = 6, evaluate
23(x − y)
y + x y + 2x
7. If a = 4, b = 3, c = 5 and d = 6, evaluate
3a + 2b
3c − 2d
8. Simplify 2x ÷ 14x y.
9. Simplify
25x
2 y z
3
5x yz
10. Multiply 3a − b by a + b.
11. Multiply 2a − 5b + c by 3a + b.
12. Simplify 3a ÷ 9ab.
13. Simplify 4a 2 b ÷ 2a.
14. Divide 6x 2 y by 2x y.
15. Divide 2x 2 + x y − y 2 by x + y.
16. Divide 3 p 2 − pq − 2q 2 by p − 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) a m ×a n = a m + n
For example, a 3 × a 4 = a 3+4 = a 7