Re call basic operations in mathematics

1. In our daily life there are four basic operations.
(a) Addition
(b) Subtraction
(c) Multiplication
(d) Division
Addition
Example-1 4 + 6 = 10
Example-2 −2 − 3 − 5 = −10
Exercise # 1
(i) Add 4 and 8
(ii) Add -4 and -8
(iii) Add -4 and 8
(iv) Add 4 and -8
Subtraction
Example- 3 8 – 5 = 3
Example-4 5 – 8 = - 3
Exercise # 2
(i) Subtract 4 from 8
(ii) Subtract -4 from -8
(iii) Subtract -4 from 8
(iv) Subtract 4 from -8

2. Multiplication
If the two real numbers being multiplied have the same sign, their product is positive.
Example- 5 (8) × (5) = 40
Example-6 (- 8) × (- 5) = 40
Exercise # 3
(i) (4) × (8)
(ii) (- 4) × (- 8)
(iii) (- 4) × (- 6)
(iv) ( 4) × ( 6)
If the two real numbers being multiplied have opposite signs, their product is negative.
Example- 7 (- 8) × (5) = - 40
Example-8 ( 8) × (- 5) = - 40
Exercise # 4
(i) (- 4) × (8)
(ii) ( 4) × (- 8)
(iii) ( 4) × (- 6)
(iv) ( - 4) × ( 6)

3. Division
If the two numbers have the same sign their quotient is positive.
Example- 9 10 ÷ 2 = 5
Example-10 ( - 10) ÷ (- 2) = 5
Exercise # 5
(i) 10 ÷ 5
(ii) 12 ÷ 5
(iii) ( - 10) ÷ (- 5)
(iv) ( - 12) ÷ (- 5)
If the two numbers have oposite signs their quotient is negative.
Example- 11 (- 10) ÷ 2 = - 5
Example-12 ( 10) ÷ (- 2) = - 5
Exercise # 6
(i) (- 10) ÷ 5
(ii) (- 12) ÷ 5
(iii) ( 10) ÷ (- 5)
(iv) ( 12) ÷ (- 5)

4. BODMAS RULE
BODMAS is stands for Bracket, Order, Division, Multiplication, Addition, and Subtraction.
Example-13 Simplify the following
2 + 3 − 5
Solution 2 + 3 − 5
= 5 − 5
= 0
Example-14 Simplify the following
2 + 3 × 2 − 5
Solution 2 + 3 × 2 − 5
= 2 + 6 − 5
= 8 − 5
= 3
Example-15 Simplify the following
7 + (10 − 2 × 3)
Solution 7 + (10 − 2 × 3)
= 7 + (10 − 6)
= 7 + 4
= 11

5. Example-16 Simplify the following
4 × 6 − 3 − 5
Solution 4 × 6 − 3 − 5
= 24 − 8
= 16
Example-17 Simplify the following
24 ÷ (8 − 5) + 7 ×7
Solution 24 ÷ (8 − 5) + 7 × 7
24 ÷ 3 + 7 × 7
= 8 + 49
= 57
Example-18 Simplify the following
4 + 9 × (7 − 5) × 2
Solution 4 + 9 × (7 − 5) × 2
4 + 9 × 2 × 2
= 4 + 36
= 40
Example-19 Simplify the following
6 ÷ [3{6 − (9 − 5)}]
Solution 6 ÷ [3{6 − (9 − 5)}]
6 ÷ [3{6 − 4}]

6. 6 ÷ [3 × 2]
6 ÷ 6
= 1
Exercise # 7
(i) 20 + 4 × 5
(ii) (9 + 10) ÷ 19
(iii) 19 ÷ (21 − 2)
(iv) 25 × 2 − 4 + 10
(v) 5 + 20 ÷ 4 × 3
(vi) 25 × 2 − (4 − 10)
(vii) 3 × (12 − 1) ÷ 11
(viii) 3 + (12 − 1) ÷ 11
(ix) (12 × 2) ÷ 12 + 2
(x) 12 − (3 × 2 − 5) − 7
(xi) 3 + 5 − (7 × 2) ÷ 2 − 1
(xii) 10 − {5 + (30 − 10) ÷ 10}
Coefficients, Base and Exponents
Example-20 4𝑥2
Where 4 is coefficient of x
x is a base
2 is exponent

7. Exercise # 8
Find coefficient, base and exponent.
(i) −3𝑥4
(ii) 9𝑥−5
(iii) 5𝑦−1
(iv) – 𝑥
Algebraic expression
Polynomial
An expression of the form
𝑎0 𝑥 𝑛
+ 𝑎1 𝑥 𝑛−1
+ 𝑎2 𝑥 𝑛−2
+ ⋯ + 𝑎 𝑛−1 𝑥 + 𝑎 𝑛
Where n is a positive integer or zero, 𝑎 𝑛 ≠ 0 and the coefficients 𝑎0, 𝑎1, 𝑎2, … . . , 𝑎 𝑛
are real numbers, is called a polynomial of degree n in one variale x.
 If we put n = 1 in above expression, then polynomial is called linear or degree 1
for example 3𝑥 − 10.
 If we put n = 2 in above expression, then polynomial is called quadratic or degree 2
for example 4𝑥2
+ 3𝑥 − 10.
 If we put n = 3 in above expression, then polynomial is called cubic or degree 3
for example 6𝑥3
+ 4𝑥2
+ 3𝑥 + 10.
Example-21 Express the polynomial 6𝑥3
+ 10 + 4𝑥2
+ 3𝑥 in ascending and
descending order.
10 + 3𝑥+4𝑥2
+ 6𝑥3
(Ascending order)
6𝑥3
+ 4𝑥2
+ 3𝑥 + 10 (Descending order)

8. Exercise # 9
Express the following polynomials in ascending order.
(i) 5𝑥3
+ 7𝑥 + 10 − 8𝑥2
(ii) −5𝑥 + 5𝑥6
+ 8𝑥3
− 7𝑥2
+ 5
(iii) 8𝑦5
− 4𝑦 − 5𝑦2
− 5𝑦3
+ 2𝑦4
(iv) 3𝑎3
+ 4𝑎 − 5𝑎2
+ 2𝑎5
− 100
Exercise # 10
Simplify the following
(i) (x3)2
(ii) (−x3)2
(iii) (x3)3
(iv) (−x3)3
(v) (2x5)2
(vi) (−2x5)2
(vii) x5
× x2
(viii) x5
÷ x2
(ix) (−x2
y3
z)4
(x) (−2x2
y3
z)4
(xi) 𝑥−2
(xii) 𝑥−1
Re-call the following
(i) ( 𝑎)2
− ( 𝑏)2
= ( 𝑎 + 𝑏)( 𝑎 − 𝑏)
(ii) ( 𝑎 + 𝑏)2
= ( 𝑎)2
+ 2( 𝑎)( 𝑏) + ( 𝑏)2
(iii) ( 𝑎 − 𝑏)2
= ( 𝑎)2
− 2( 𝑎)( 𝑏) + ( 𝑏)2
(iv) ( 𝑎 + 𝑏)3
= ( 𝑎)3
+ 3( 𝑎)2( 𝑏) + 3( 𝑎)( 𝑏)2
+ ( 𝑏)3
(v) ( 𝑎 − 𝑏)3
= ( 𝑎)3
− 3( 𝑎)2( 𝑏) + 3( 𝑎)( 𝑏)2
− ( 𝑏)3

9. We can also perform four basic operations on algebraic expressions
(a) Addition
(b) Subtraction
(c) Multiplication
(d) Division
Addition
Example-22 𝐴𝑑𝑑 4𝑥 𝑡𝑜 10
4𝑥 + 10
Example-23 𝐴𝑑𝑑 4𝑥 𝑡𝑜 2𝑥
4𝑥 + 2𝑥 = 6𝑥
Example-24 𝐴𝑑𝑑 3𝑥 𝑡𝑜 4𝑦
3𝑥 + 4𝑦
Example-25 (4𝑥 + 3𝑦) 𝑡𝑜(6𝑥 − 𝑦)
= (4𝑥 + 3𝑦) + (6𝑥 − 𝑦)
= 4𝑥 + 3𝑦 + 6𝑥 − 𝑦
= 10𝑥 + 2𝑦
Exercise # 11
(i) 𝐴𝑑𝑑 ( 𝑥 + 4) 𝑡𝑜 4𝑥
(ii) 𝐴𝑑𝑑 ( 𝑥 + 4) 𝑡𝑜 ( 𝑥 − 4)
(iii) 𝐴𝑑𝑑(3𝑥2
+ 𝑥) 𝑡𝑜 ( 𝑥2
+ 3𝑥)
(iv) 𝐴𝑑𝑑( 𝑥2
+ 3𝑦 − 2) 𝑡𝑜 ( 𝑥2
− 4𝑦 + 10)
(v) 𝐴𝑑𝑑(2𝑥2
𝑦 − 3𝑥𝑦 − 𝑥𝑦2) 𝑡𝑜 (−3𝑥𝑦2
+ 5𝑥𝑦 − 𝑥2
𝑦)

10. Subtraction
Example-26 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡 10 𝑓𝑟𝑜𝑚 4𝑥
4𝑥 − 10
Example-27 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡 2𝑥 𝑓𝑟𝑜𝑚 4𝑥
= 4𝑥 − 2𝑥
= 2𝑥
Example-28 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡 2𝑥 𝑓𝑟𝑜𝑚 4𝑦
= 4𝑦 − 2𝑥
Example-29 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡 (6𝑥 − 𝑦) 𝑓𝑟𝑜𝑚 (4𝑥 + 3𝑦)
= (4𝑥 + 3𝑦) − (6𝑥 − 𝑦)
= 4𝑥 + 3𝑦 − 6𝑥 + 𝑦
= −2𝑥 + 4𝑦
Exercise # 12
(i) Subtract (x + 4) from 4x
(ii) Subtract (x + 4) from (x − 4)
(iii) Subtract(3x2
+ x) from (x2
+ 3x)
(iv) Subtract(x2
+ 3y − 2) from (x2
− 4y + 10)
(v) Subtract(2x2
y − 3xy − xy2)from (−3xy2
+ 5xy − x2
y)

11. Multiplication
Example-30 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 4𝑥 𝑏𝑦 10
= 10(4𝑥)
= 40𝑥
Example-31 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 4𝑥 𝑏𝑦 2𝑥
= 2𝑥(4𝑥)
= 8𝑥2
Example-32 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 (3𝑥 + 4𝑦) 𝑏𝑦 3
= 3(3𝑥 + 4𝑦)
= 9𝑥 + 12𝑦
Example-33 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 (3𝑥 + 4𝑦) 𝑏𝑦 3𝑥
= 3𝑥(3𝑥 + 4𝑦)
= 9𝑥2
+ 12𝑥𝑦
Example-34 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 (3𝑥 + 4𝑦) 𝑏𝑦 (2𝑥 + 5𝑦)
= (2𝑥 + 5𝑦)(3𝑥 + 4𝑦)
= 2𝑥(3𝑥 + 4𝑦) + 5𝑦(3𝑥 + 4𝑦)
= 6𝑥2
+ 8𝑥𝑦 + 15𝑥𝑦 + 20𝑦2
= 6𝑥2
+ 23𝑥𝑦 + 20𝑦2

12. Example-35 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 (3𝑥 − 4𝑦) 𝑏𝑦 (2𝑥 + 5𝑦)
= (2𝑥 + 5𝑦)(3𝑥 − 4𝑦)
= 2𝑥(3𝑥 − 4𝑦) + 5𝑦(3𝑥 − 4𝑦)
= 6𝑥2
− 8𝑥𝑦 + 15𝑥𝑦 − 20𝑦2
= 6𝑥2
+ 7𝑥𝑦 − 20𝑦2
Exercise # 13
(i) 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑥2
𝑏𝑦 2
(ii) 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑥2
𝑏𝑦 2𝑥
(iii) 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑥2
𝑏𝑦 𝑦
(iv) 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 12𝑥4
𝑦 𝑏𝑦6𝑥5
(v) 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 (6𝑥 − 10𝑥𝑦) 𝑏𝑦 4𝑥𝑦
(vi) 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 (3𝑥 − 2𝑦) 𝑏𝑦 ( 𝑥 − 𝑦)
Division
Example-36 𝐷𝑖𝑣𝑖𝑑𝑒
4𝑥 𝑏𝑦 10
=
4𝑥
10
=
2𝑥
5
Example-37 𝐷𝑖𝑣𝑖𝑑𝑒 4𝑥2
𝑏𝑦 2𝑥
=
4𝑥2
2𝑥
= 2𝑥

13. Example-38 𝐷𝑖𝑣𝑖𝑑𝑒 2𝑥 𝑏𝑦 4𝑥2
=
2𝑥
4𝑥2
=
1
2𝑥
Example-39 𝐷𝑖𝑣𝑖𝑑𝑒 ( 𝑎 + 𝑏)4
𝑏𝑦 ( 𝑎 + 𝑏)
=
( 𝑎 + 𝑏)4
( 𝑎 + 𝑏)
= ( 𝑎 + 𝑏)3
Example-40 𝐷𝑖𝑣𝑖𝑑𝑒 ( 𝑎 + 𝑏) 𝑏𝑦 ( 𝑎 + 𝑏)4
=
( 𝑎 + 𝑏)
( 𝑎 + 𝑏)4
=
1
( 𝑎 + 𝑏)3
Exercise # 14
(i)
6𝑥5
12𝑥4 𝑦
(iv) 𝐷𝑖𝑣𝑖𝑑𝑒
(𝑥−4)
𝑥+𝑦
𝑏𝑦
2
(𝑥+𝑦)
(ii)
16𝑥3 𝑦4
24𝑥5 𝑦2 (v) 𝐷𝑖𝑣𝑖𝑑𝑒 4𝑥𝑦 𝑏𝑦 (6𝑥 − 10𝑥𝑦)
(iii)
15𝑦𝑧4
25𝑥2 𝑦𝑧
(vi) 𝐷𝑖𝑣𝑖𝑑𝑒 4( 𝑥 + 𝑦)3
𝑏𝑦 2( 𝑥 + 𝑦)

14. Addition and subtraction of fractions
Example-41
2
5
+
1
5
=
2+1
5
=
3
5
Example-42
2
5
−
1
5
=
2−1
5
=
1
5
Example-43
2
5
+
1
3
=
3(2)+5(1)
15
=
6+5
15
=
11
15
Example-44
2
5
−
1
3
=
3(2)−5(1)
15
=
6−5
15
=
1
15

15. Example-45
3
8
+
5
6
=
3(3)+5(4)
24
=
9+20
24
=
29
24
Example-46
3
12
+
5
16
=
3(4)+5(3)
48
=
12+15
48
=
27
48
Example-47
𝑥
4
−
𝑥−3
6
=
3(𝑥)−2(𝑥−3)
12
=
3𝑥−2𝑥+6
12
=
𝑥+6
12
Example-48
𝑥+5
7
+
𝑥−3
4
=
4(𝑥+5)+7(𝑥−3)
28
=
4𝑥+20+7𝑥−21
28
=
11𝑥−1
28

16. Answers
Exercise # 1
(i) 12 (ii) -12 (iii) 4 (iv) -4
Exercise # 2
(i) 4 (ii) -4 (iii) 12 (iv) -12
Exercise # 3
(i) 32 (ii) 32 (iii) 24 (iv) 24
Exercise # 4
(i) -32 (ii) -32 (iii) -24 (iv) -24
Exercise # 5
(i) 2 (ii) 2.4 (iii) 2 (iv) 2.4
Exercise # 6
(i) -2 (ii) -2.4 (iii) -2 (iv) -2.4
Exercise # 7
(i) 40 (ii) 1 (iii) 1 (iv) 56 (v) 20 (vi) 56
(vii) 3 (viii) 4 (ix) 4 (x) 4 (xi) 0 (xii) 3
Exercise # 8
(i) -3, x, 4 (ii) 9, x, -5 (iii) 5, y, -1 (iv) -1, u, 1
Exercise # 9
(i) 10 + 7𝑥 − 8𝑥2
+5𝑥3
(ii) 5 − 5𝑥 − 7𝑥2
+ 8𝑥3
+ 5𝑥6
(iii) −4𝑦 − 5𝑦2
− 5𝑦3
+ 2𝑦4
+ 8𝑦5
(iv) −100 + 4𝑎 − 5𝑎2
+ 3𝑎3
+ 2𝑎5

17. Exercise # 10
(i) 𝑥6
(ii) 𝑥6
(iii) 𝑥9
(iv) −𝑥−9
(v) 4𝑥10
(vi) 4𝑥10
(vii) 𝑥7
(viii) 𝑥3
(ix) 𝑥8
𝑦12
𝑧4
(x) 16𝑥8
𝑦12
𝑧4
(xi)
1
𝑥2
(xii)
1
𝑥
Exercise # 11
(i) 5𝑥 + 4
(ii) 2𝑥
(iii) 4𝑥2
+ 4𝑥
(iv) 2𝑥2
− 𝑦 + 8
(v) 𝑥2
𝑦 + 2𝑥𝑦 − 4𝑥𝑦2
Exercise # 12
(i) 3x − 4
(ii) −8
(iii) −2x2
+ 2x
(iv) −7y + 12
(v) −2xy2
+ 8xy − 3x2
y
Exercise # 13
(i) 2𝑥2
(ii) 2𝑥3
(iii) 𝑥2
𝑦
(iv) 72𝑥9
𝑦
(v) 24𝑥2
𝑦 − 40𝑥2
𝑦2
(vi) 3𝑥2
− 5𝑥𝑦 + 2𝑦2
Exercise # 14
(i)
𝑥
2𝑦
(iv)
(𝑥−4)
2
(ii)
2𝑦2
3𝑥2
(v)
2𝑦
(3−5𝑦)
(iii)
3𝑧3
5𝑥2
(vi) 2( 𝑥 + 𝑦)2