Algebric Expression:
The word algebra comes from an Arabic word (al-jabr, literally means restoration). Its roots can be traced to a mathematician, Muhammed bin Musa al-Khwarizmi (780 - 850).
He wrote the Compendious Book on Calculation by Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic.
Q. What is algebraic expressions?
Ans: Meaningful organization of operational signs and numerical letter symbols is called Algebraic Expressions.
Such as, 2a + 3b - 4c is an algebraic expressions.In algebraic expression, different types of information are expressed through the letters a,b,c,p,q, m,n,x,y,z.........etc.These alphabet are used to solve different types of problems related to algebraic expressions.
In arithmetic, only positive numbers are used, where as, in algebra, both positive and negative numbers including 0 are used.Algebra is the generalization of arithmetic.The numbers used in algebraic expressions are constants, their values are fixed.The letter symbol used in algebraic expressions are constants, their values are fixed.The letter symbol used in algebraic expressions are variables, their values are not fixed, they can be of any value.
Q. What is algebraic formula?
Ans: Any general rule or resolution expressed by algebraic symbols is celled Algebraic formula.Formula:(a+b)² = a² + 2ab + b²(a - b)² = a² - 2ab + b²
Corollary:
1. a² + b² ＝（a + b)² ─ 2ab
2. a² + b² ＝(a- b)² + 2ab
3. (a + b)² ＝(aーb)² + 4ab
4. (a ─ b)² ＝(a + b)² - 4ab
5. a² - b² = （a + b) (a- b)
Formula-4:
(x + a) (x + b) = x² + ( a +b ) x + ab
Formula-5:
(a + b +c)² = a² + b² + c² + 2ab + 2bc + 2ac
Corollary - 7:
a² + b² + c² = (a + b +c)² - 2( ab +bc + ca)
Corollary - 8:
2(ab + bc + ac ) = (a + b +c) ² - ( a² + b² + c²)
Ex.1. What is the square of (4x + 5y) ?
Solution:
( 4x + 5y) = (4x )² + 2. (4x) . (5y) + (5y)²
= 16x² + 40 xy + 25 y²
Ex.2. What is the square of (3a - 7b) ?
(3a - 7b)²
= (3a) - 2.3a.7b + (7b)²
= 9a² - 42ab + 49b²
Ex. 3. Find the square of 996 by applying the formula of square.
(996) = (1000 - 4)
= (1000)² - 2.1000.4 + 4²
= 992016
4. What is the square of a+b+c+d?
(a + b + c + d)²
= { (a + b) + ( c + d) }²
= ( a + b)² + 2 ( a + b) (c + d) + ( c + d)²
= (a + b)² + 2 ( a +b) (c + d) + ( c + d)²
= a² + 2ab + b² + 2(ac + ad + bc + bd) + c² + 2cd + d²
= a² + 2ab + b² + 2ac + 2ad +2bc +2bd + c² + 2cd + d²
Ex.5. Simplify:
(5x + 7y + 3z)² + 2 (7x - 7y - 3z) (5x + 7y + 3z) + ( 7x - 7y - 3z)²
Let, 5x + 7y + 3z = a
5x + 7y + 3z = b
Given expression = a² + 2ab + b²
= (a + b)²
= (5x + 7y + 3z + 7x - 7y - 3z)²
Substituting the value of a and b
= (12x)²
= 144x² Ans.
Ex.6. If x - y = 2 and xy = 24,
What is the value of x + y?
Solution:
(x + y)² = (x - y)² + 4xy
= 2² + 4 x 24
= 100 Ans.
Ex. 7. If a⁴ + a²b² + b⁴ = 3
a² + ab + b² = 3
What is the value of a² + b² ?
Solution:
a⁴ + a²b² + b⁴
= (a²)² + 2a²b² + (b²)² - a²b²
= (a² + b²) - (ab)²
= (a² + ab + b²) ( a² - ab + b²)
∴ = 3 = 3 (a² - ab + b² )
( Substituting the values)
or a² - ab + b² = 3/3 = 1
Now adding , a² + ab + b² = 3
a² - ab + b² = 1
We get, 2 (a² + b²) = 4
or, a² + b² = 2
∴ a² + b² = 2
Long Division in Algebra
The degree of an expression in x is the degree of the highest power of x it contains
the degree of 2x³ - 4x² + 6x - 1 is 3.
When the expression is arranged so that the highest power of x comes first, the next highest power of x second and so on, it is said to be in descending powers of x.
When the powers of x increase, the expression is said to be arranged in ascending powers of x.
Ex. Divide 2x² + 7x + 6 by x + 2.
2x + 3
x + 2) 2x² + 7x + 6 (
2x² + 4x
__________
3x + 6
3x + 6
________
The quotient is 2x + 3.
Ex. 2 . Divide 4x⁴ - 3x³ + 2x² - 5x + 6 by x² - 3x - 1.
4x² - 9x - 33
x² - 3x - 1 ) 4x⁴ - 3x³ + 2x² - 5x + 6 (
4x⁴ - 12x³ + 4x²
__________________
9 x³ + 6x² - 5x
9 x³ + 27x² - 9x
____________________
33x² - 4x + 6
33x² - 99x-33
____________________
103 x + 39
The quotient is 4x² - 9x - 33;
The remainder is 103 x + 39.
Solving Linear Equations:
Finding an Unknown in a Formula:
a) V = ibh
When l = 5, b = 2, h = 3,
V = 5 x 2 x 3,
V = 30 cm³
Volume of the cuboid = 30 cm³
b) V = lbh
When V = 240, b = 6, h = 5,
l x 6 x 5 = 240
30l = 240
l = 240/30
= 8 cm
Length of the cuboid = 8 cm
Ex. If y + b = ay + c/b calculate the value of c when y = 12,
b = 3 and a = 14.
Solution:
y + b = ay + c/b
When y = 12, b=3, a = 14,
12 + 3 = 14 x 12 + c/3
15 = 168 + c/3
3 x 15 = 168 + c / 3
45 = 168 + c
45 - 168 = c
c = 123
Construction of a Formula
Ex. i) Find a formula for the sum S of any three consecutive even numbers.Solution:
Let the smallest even number be n.
The next even number will be n + 2.
The greatest even number will be (n + 2) + 2 = n + 4
S = n + (n + 2 ) + (n + 4)
= n + n + n + 2 + 4
= 3n + 6