Khullakitab.comClass 9 Mathematics Note
ALGEBRAIC EXPRESSIONS
Algebraic expressions
Factorization
In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original.
Some important formulas
a2 - b2 = (a-b)(a+b)
Difference of Cubes
a3 - b3 = (a - b)(a2+ ab + b2)
Sum of Cubes
a3 + b3 = (a + b)(a2 - ab + b2)
Formula for (a+b)2 and (a-b)3
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab +b2
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a - b)3 = a3 - 3a2b + 3ab2 - b3
Highest common Factor and lowest common multiple
The highest common factor is calculated by multiplying all the factors which appear in both lists: So the HCF of 60 and 72 is 2 × 2 × 3 which is 12. The lowest common multiple is calculated by multiplying all the factors which appear in either list: So the LCM of 60 and 72 is 2 × 2 × 2 × 3 × 3 × 5 which is 360.
Examples
1.Resolve into factors
2a(a-1) –a +1
Soln:
= 2a(a – 1) – a + 1.
= 2a(a – 1) – (a – 1)
= (a – 1)(2a – 1)
mx2 + my2 – nx2 – ny2
Soln:
mx2 + my2 – nx2 – ny2
= m(x2 + y2) – n(x2 + y2)
= (x2 + y2)(m – n)
2.Factorize
64x6 – y6.
= (8x3)2 – (y3)2
= (8x3 + y3)(8x3 – y3)
= {(2x)3 + (y)3}{(2x)3 – (y)3}
= (2x + y)(4x2 – 2x + y2)(2x – y)(4x2 + 2xy + y2).
= (2x + y)(2x – y)(4x2 – 2x + y2)(4x2 + 2xy + y2).
Find the H.C.F and L.C.M of the following expressions
1st expression = a2 +2ab + b2
= (a+b)(a+b)
2nd expression = b2 -a2 +2bc +c2
=b2 +2bc +c2 -a2
= (b+c)2 - a2
= (b + c + a ) (b + c- a )
3rd expression = - b2+ a2 +2ca +c2
=(a+c)2- b2
= (a+c-b)(a+c+ b)
H.C.F = 1
L.C.M =(a+b)(a+b)(b + c + a )(b + c- a )(a+c-b)
3.Simplify
64x6 – y6.
= (8x3)2 – (y3)2
= (8x3 + y3)(8x3 – y3)
= {(2x)3 + (y)3}{(2x)3 – (y)3}
= (2x + y)(4x2 – 2x + y2)(2x – y)(4x2 + 2xy + y2).
= (2x + y)(2x – y)(4x2 – 2x + y2)(4x2 + 2xy + y2).
x3y – 64y4
= y(x3 – 64y3)
= y{(x)3 – (4y)3}
= y(x – 4y)(x2 + 4y + 16y2)