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Exploring Algebraic Identities – Class 9 Mathematics notes, important questions and MCQs

NCERT Class 9 Mathematics, chapter 4. Notes, key terms, important questions with answers and MCQs.

Chapter overview

Algebraic identities are equations true for all values of the variables. They make expansion, factorisation and quick calculation easy.

Basic identities

(a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b².
a² − b² = (a + b)(a − b).
(x + a)(x + b) = x² + (a + b)x + ab.

Three-term and cubic identities

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
(a + b)³ = a³ + b³ + 3ab(a + b); (a − b)³ = a³ − b³ − 3ab(a − b).
a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).

Uses

Mental calculation: 102² = (100 + 2)² = 10404; 99 × 101 = 100² − 1 = 9999.
Factorising: x² − 9 = (x + 3)(x − 3); 8x³ + 27 = (2x + 3)(4x² − 6x + 9).
If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Key terms

Identity: an equation true for all values of its variables
Expansion: writing a product as a sum of terms
Factorisation: writing an expression as a product of factors

Formulas

(a + b)² = a² + 2ab + b²
a² − b² = (a + b)(a − b)
(a + b)³ = a³ + b³ + 3ab(a + b)
a³ + b³ = (a + b)(a² − ab + b²)

Important questions with answers

Evaluate 103 × 97 using an identity.2 marks

(100 + 3)(100 − 3) = 100² − 3² = 10000 − 9 = 9991.

Factorise 4x² + 12xy + 9y².3 marks

It is (2x)² + 2(2x)(3y) + (3y)² = (2x + 3y)².

If x + y = 12 and xy = 27, find x² + y² and x³ + y³. Also verify using actual values of x and y.5 marks

x² + y² = (x + y)² − 2xy = 144 − 54 = 90. x³ + y³ = (x + y)³ − 3xy(x + y) = 1728 − 3 × 27 × 12 = 1728 − 972 = 756. Check: x and y are 9 and 3; 81 + 9 = 90 ✓ and 729 + 27 = 756 ✓.

Quick revision

Square of a sum has the middle term 2ab.
Difference of squares factorises into (a + b)(a − b).
a + b + c = 0 ⇒ a³ + b³ + c³ = 3abc.

Exam tips

Look for a² − b² first when factorising.
Check an identity by putting a simple value like a = 1, b = 2.

Practice MCQs

(x + 3)² equals
1. x² + 9
2. x² + 6x + 9
3. x² + 3x + 9
4. x² − 6x + 9
Show answer
Answer: (b) x² + 6x + 9. a² + 2ab + b².
51² − 49² equals
1. 100
2. 4
3. 200
4. 2
Show answer
Answer: (c) 200. (51 + 49)(51 − 49) = 200.
x² − 16 factorises as
1. (x − 4)²
2. (x + 4)(x − 4)
3. (x − 8)(x + 2)
4. (x + 16)(x − 1)
Show answer
Answer: (b) (x + 4)(x − 4). Difference of squares.
103 × 97 equals
1. 9981
2. 9999
3. 10009
4. 9991
Show answer
Answer: (d) 9991. (100 + 3)(100 − 3) = 10000 − 9 = 9991.
(x + 2)(x + 5) equals
1. x² + 3x + 10
2. x² + 10x + 7
3. x² + 7x + 10
4. x² + 7x + 7
Show answer
Answer: (c) x² + 7x + 10. x² + (2 + 5)x + 2 × 5.
If x + 1/x = 4, then x² + 1/x² =
1. 14
2. 16
3. 18
4. 12
Show answer
Answer: (a) 14. (x + 1/x)² − 2 = 16 − 2 = 14.
Without cubing, (−12)³ + 7³ + 5³ equals
1. −1260
2. 1260
3. 0
4. −420
Show answer
Answer: (a) −1260. Sum is 0, so value = 3abc = 3(−12)(7)(5) = −1260.
27x³ − 8 factorises as
1. (3x + 2)(9x² − 6x + 4)
2. (3x − 2)³
3. (3x − 2)(9x² + 6x + 4)
4. (9x − 4)(3x + 2)
Show answer
Answer: (c) (3x − 2)(9x² + 6x + 4). a³ − b³ = (a − b)(a² + ab + b²).

All chapters of Class 9 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.