Algebraic identities: Overview, Questions, Preparation

Based on the number of terms i.e. one, two, or three that are present in algebraic expressions, they are referred to as monomials, binomials, and trinomials, respectively. And, when an expression consists of one or more terms, then it is termed as a polynomial. The term of an algebraic expression has a number attached to it, which is known as a coefficient. 

Standard Algebraic Identities List

1. (a+b)²=a²+2ab+b²

2. (a-b)²=a²-2ab+b²

3. (a+b) (a-b) = a²- b²

If in any of the above three expressions, values are put for terms a and b, the left-hand side of the equation will be equal to the right-hand side of the equation. Thus, proving these expressions as Identities. 

Illustration 1: To Prove the Third Standard Algebraic Identity 
(a+b) (a-b) = a² - b²
Proof: Starting from the Left Hand Side, using Distributive Law
(a+b) (a-b) = a(a-b) + b(a-b)
By multiplying each of the terms, we get the following,
(a+b) (a-b) = a² -ab +ab -b²
(a+b) (a-b) = a² - b²
 L.H.S. = R.H.S.

Important Topics Covered Under this Chapter

• Meaning of Expressions
• What are Terms, Factors, and Coefficients
• What are Monomials, Binomials, and Polynomials
• Like and Unlike Terms
• Addition and Subtraction of Algebraic Expressions 
• Multiplication of Algebraic Expressions
• Monomial by a Monomial
• Monomial by a Polynomial
• Polynomial by a Polynomial
• Meaning of an Identity
• What are the Standard Identities
• Application of Identities 

Illustration 2: Adding of Algebraic Expressions
   4p²q² - 5pq + 6, 7 + 9pq - 5p²q²
= (4p²q² - 5pq + 6) + (7 + 9pq - 5p²q²)
= 4p²q²  - 5p²q² - 5pq + 9pq + 6 + 7
= -p²q²+ 4pq + 13

Illustration 3: Subtraction of Algebraic Expressions
   5xy + 7yz - 9zx from 7xy -4yz - 4zx + 12xyz
= (7xy - 4yz - 4zx + 12xyz) - (5xy + 7yz - 9zx)
= 7xy - 5xy - 4yz - 7yz - 4zx + 9zx + 12xyz
= 2xy - 11yz + 5zx + 12xyz

Illustration 4: Multiplication of Algebraic Expressions
   (7 - 4x) (5 + x)
= 7 (5 + x) - 4x (5 + x)
= 35 + 7x - 20x - 4x²
= 35 - 13x - 4x² 

Illustration 5: Simplification
   (x² - 5) (x + 5) + 25
= x² (x + 5) - 5 (x + 5) +25
= x³ + 5x² - 5x - 25 + 25
=x³ + 5x² - 5x

Illustration 6: Show that L.H.S. = R.H.S.
   (5x + 9)² - 180x = (5x - 9)²
L.H.S. = (5x + 9)² - 180x
= 25x² + 2(5x * 9) + 81 - 180x
= 25x² + 90x + 81 - 180x
= 25x² - 90x + 180x
= (5x - 9)²
= R.H.S. 

For class 9, under chapter Algebraic Identities the concept of L.H.S. = R.H.S. has been elaborated. In class 9, students will come to discover the algebraic identities of all the algebraic expressions and formulas. The algebraic formulas taught in class 9 are the mathematical rules that are expressed using symbols. But, with the help of algebraic identities, it is shown that the equation stands true for all the values of the variables.Overall, the Algebra section in Class 8 and 9 question papers is of total 17 marks.

Algebraic Identities for class 9 formulas with two variables i.e. X and Y

1. (x+y)²=x²+2xy+y²

2. (x-y)²=x²-2xy+y²

3. x²- y²= (x+y) (x-y)

4.(x+a) (x+b) = x² +(a+b)x+ab;a and b are two constant values

5. (x+y)³= x³+y³+ 3xy(x+y)

6. (x-y)³= x³-y³+ 3xy(x-y)

Algebraic Identities for class 9 formulas with three variables i.e. X, Y, and Z

(x+y+z)²=x²+y²+z²+2xy+2yz+2xz

x³+y³+z³-3xyz=(x+y+z) (x²+y²+z²+-xy-yz-zx)

Illustration 7: Using the appropriate above given algebraic Identities solve,
   (x  + 6) (x - 6)
Using the algebraic identity, x² - y² = (x + y) (x - y), the given expression can be written as:
= (x + 6) (x - 6) = x² - 6² 
= x² - 36

Q: What do Algebraic Identities mean?

Q: State the number of identities that are there for class the 8 Algebraic Expressions chapter?   

Q: What is the XAXB formula in the Algebraic Identities chapter?

Q: Practicing every exercise given in the NCERT class 8 and 9 textbooks for the chapter Algebraic Expressions and Identities is necessary for the examinations? 

Q: What do ‘Terms’ mean in algebra?