These Class 9 Maths Notes and Chapter 4 Exploring Algebraic Identities Class 9 Ganita Manjari Notes are designed according to the latest CBSE syllabus.
Class 9 Maths Chapter 4 Exploring Algebraic Identities Notes
Class 9 Maths Ganita Manjari Chapter 4 Notes
Class 9 Exploring Algebraic Identities Notes
→ Introduction: Algebraic Identities are the equations that are true for all values of the variable(s) occurring in it. For example, 2x + 1 = 2(x + 1) – 1 is an identity, as it is true for any real value of variable x. If we put x = 0 in the LHS and the RHS of the above equation, we get their values same.
At x = 0,LHS = 2 × 0 + 1 = 0+ 1 = 1 and RHS = 2(0 + 1) – 1 = 2(1) – 1 = 2 – 1 = 1.
So, we get LHS = RHS for x = 0.
We can check that for x = 1,
LHS = 2 × 1 + 1 = 2 + 1 = 3 and RHS = 2(1 + 1) – 1 = 2 × (2) – 1 = 4 – 1 = 3.
So, we get LHS = RHS.
This is true for any other real values of the variable x as 2x + 1 = 2(x + 1) – 1 is an algebraic identity.
On the other hand, an algebraic equation is true for a few values of variable(s).
For example, 2x – 1 = 0 is true for x = \(\frac{1}{2}\) only and for no other value of the variable.
A beautiful thing about identities is that they can be visualised by using geometrical models or algebra tiles.
→ Some Frequently Used Useful Identities:
- (x + y)2 = x2 + 2xy + y2
- (x – y)2 = x2 – 2xy + y2
- (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
- (x + y)(x – y) = x2 – y2
- (x + a)(x + b) = x2 + (a + b)x + ab
- (ax + b)(cx + d) = acx2 + (ad + bc)x + bd
- x3 – y3 = (x – y)(x2 + xy + y2)
- x3 + y3 = (x + y)(x2 – xy + y2)
- (x + y)3 = x3 + 3x2y + 3xy2 + y3 OR, (x + y)3 = x3 + y3 + 3xy(x + y)
- (x – y)3 = x3 – 3x2y + 3xy2 – x3 OR, (x + y)3 = x3 – y3 – 3xy(x – y)
- x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz – zx)
If x + y + 2 = 0, then x3 + y3 + z3 = 3xyz.
→ Uses of Algebraic Identities:
(i) They can be used to factorise the algebraic expressions.
For example, 9a2 + 24ab + 16b2 can be factorised using the identity (x + y)2 = x2 + 2xy + y2.
We can write the given algebraic expression as:
9a2 + 24ab + 16b2 = (3a)2 + 2(3a)(4b) + (4b)2 = (3a + 4b)2 = (3a + 4b) (3a+ 4b).
(ii) They can be used to simplify the calculations while calculating squares and cubes of numbers. For example, we can use identity to simplify the calculation of (102)2.
We can write (102)2 = (100 + 2)2
= 1002 + 2 × 100 × 2 + 22 [using (x + y)2 = x2 + 2xy + y2]
= 10000 + 400 + 4 = 10404.
(iii) Rational algebraic expressions may be simplified by factorisation and removing the common factors in the numerator and denominator, provided such a factor exists and it is not equal to zero.
For example, \(\frac{x^2-2^2}{x-2}\), x ≠ 2 an be simplified as \(\frac{x^2-2^2}{x-2}=\frac{(x+2)(x-2)}{x-2}\) (using the identify (x – y) = (x – y)(x + y))
After cancelling the common factor (x – 2) in the numerator and denominator, we get \(\frac{x^2-2^2}{x-2}\) = (x + 2)