basic_engineering_mathematics0

Algebra 43
x 2 is a common factor of the first two terms, thus:
x 3 + 3x 2 − x − 3 = x 2 (x + 3) − x − 3
−1 is a common factor of the last two terms, thus:
Multiplication is performed before addition and subtraction thus:
2a + 5a × 3a − a = 2a + 15a 2 − a
= a + 15a 2 = a(1 + 15a)
x 2 (x + 3) − x − 3 = x 2 (x + 3) − 1(x + 3)
Problem 39.
Simplify (a + 5a) × 2a − 3a
(x + 3) is now a common factor, thus:
x 2 (x + 3) − 1(x + 3) = (x + 3)(x 2 − 1)
Now try the following exercise
The order of precedence is brackets, multiplication, then subtraction.
Hence
(a + 5a) × 2a − 3a = 6a × 2a − 3a = 12a 2 − 3a
= 3a(4a − 1)
Exercise 23
Further problems on brackets and
factorization (Answers on page 273)
Problem 40. Simplify a + 5a × (2a − 3a)
In Problems 1 to 13, remove the brackets and simplify where
possible:
1. (x + 2y) + (2x − y)
2. (4a + 3y) − (a − 2y)
3. 2(x − y) − 3(y − x)
4. 2x 2 − 3(x − xy) − x(2y − x)
5. 2(p + 3q − r) − 4(r − q + 2p) + p
6. (a + b)(a + 2b)
7. (p + q)(3p − 2q)
8. (i) (x − 2y) 2 (ii) (3a − b) 2
9. 3a(b + c) + 4c(a − b)
10. 2x + [y − (2x + y)]
11. 3a + 2[a − (3a − 2)]
12. 2 − 5[a(a − 2b) − (a − b) 2 ]
13. 24p − [2(3(5p − q) − 2(p + 2q)) + 3q]
In Problems 14 to 17, factorize:
14. (i) pb + 2pc (ii) 2q 2 + 8qn
15. (i) 21a 2 b 2 − 28ab (ii) 2xy 2 + 6x 2 y + 8x 3 y
16. (i) ay + by + a + b (ii) px + qx + py + qy
17. (i) ax − ay + bx − by (ii) 2ax + 3ay − 4bx − 6by
The order of precedence is brackets, multiplication, then subtraction.
Hence
a + 5a × (2a − 3a) = a + 5a ×−a = a +−5a 2
Problem 41.
= a − 5a 2 = a(1 − 5a)
Simplify a ÷ 5a + 2a − 3a
The order of precedence is division, then addition and subtraction.
Hence
a ÷ 5a + 2a − 3a = a + 2a − 3a
5a
Problem 42.
= 1 5 + 2a − 3a = 1 5 − a
Simplify a ÷ (5a + 2a) − 3a
The order of precedence is brackets, division and subtraction.
Hence
a ÷ (5a + 2a) − 3a = a ÷ 7a − 3a
= a 7a − 3a = 1 7 − 3a
6.4 Fundamental laws and precedence
The laws of precedence which apply to arithmetic also apply
to algebraic expressions. The order is Brackets, Of, Division,
Multiplication, Addition and Subtraction (i.e. BODMAS)
Problem 43. Simplify a ÷ (5a + 2a − 3a)
The order of precedence is brackets, then division. Hence:
a ÷ (5a + 2a − 3a) = a ÷ 4a = a 4a = 1 4
Problem 38.
Simplify 2a + 5a × 3a − a
Problem 44.
Simplify 3c + 2c × 4c + c ÷ 5c − 8c