Class 8 Maths Algebraic Expressions and Identities Notes & MCQs

Algebraic Expressions and Identities – Quick Notes Class 8

1. Algebraic Expressions

• An algebraic expression contains variables, constants, and arithmetic operations.
• Examples: $3x + 5$3x+5, $2a^2 – 3b + 7$2a2−3b+7

Terms: The parts of an expression separated by + or −.

• Example: $3x + 5y − 7$3x+5y−7 → Terms: $3x, 5y, -7$3x,5y,−7

Coefficient: The numerical factor of a term.

• Example: In $3x$3x, coefficient = 3

Types of Expressions:

1. Monomial: One term ($5x$5x)
2. Binomial: Two terms ($x + 5$x+5)
3. Trinomial: Three terms ($x + y + z$x+y+z)

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2. Algebraic Identities

• Identity: An equation true for all values of the variables.

Important Identities:

1. Square of a sum:

$$(a + b)^2 = a^2 + 2ab + b^2$$(a+b)2=a2+2ab+b2

2. Square of a difference:

$$(a – b)^2 = a^2 – 2ab + b^2$$(a−b)2=a2−2ab+b2

3. Product of sum and difference:

$$(a + b)(a – b) = a^2 – b^2$$(a+b)(a−b)=a2−b2

4. Cube of a sum:

$$(a + b)^3 = a^3 + b^3 + 3ab(a + b)$$(a+b)3=a3+b3+3ab(a+b)

5. Cube of a difference:

$$(a – b)^3 = a^3 – b^3 – 3ab(a – b)$$(a−b)3=a3−b3−3ab(a−b)

6. Sum of cubes:

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

7. Difference of cubes:

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

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Tips / Tricks:

• Always identify type of identity first before expanding or factoring.
• Use shortcut formulas to simplify calculations quickly.
• Check signs carefully when applying cube and square formulas.

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Algebraic Expressions and Identities – MCQ Q&A

1. Q: What is the coefficient of x in $5x + 7$5x+7?
   A: 5
2. Q: Number of terms in $3x + 4y − 7$3x+4y−7?
   A: 3
3. Q: Is $x + 2$x+2 a monomial, binomial, or trinomial?
   A: Binomial ✅
4. Q: Expand $(x + 3)^2$(x+3)2
   A: $x^2 + 6x + 9$x2+6x+9
5. Q: Expand $(a − b)^2$(a−b)2
   A: $a^2 − 2ab + b^2$a2−2ab+b2
6. Q: Expand $(x + 5)(x − 5)$(x+5)(x−5)
   A: $x^2 − 25$x2−25
7. Q: Expand $(a + b)^3$(a+b)3
   A: $a^3 + b^3 + 3ab(a + b)$a3+b3+3ab(a+b)
8. Q: Expand $(a − b)^3$(a−b)3
   A: $a^3 − b^3 − 3ab(a − b)$a3−b3−3ab(a−b)
9. Q: Factorize $x^2 − 16$x2−16
   A: $(x + 4)(x − 4)$(x+4)(x−4)
10. Q: Factorize $a^3 + b^3$a3+b3
    A: $(a + b)(a^2 − ab + b^2)$(a+b)(a2−ab+b2)
11. Q: Factorize $a^3 − b^3$a3−b3
    A: $(a − b)(a^2 + ab + b^2)$(a−b)(a2+ab+b2)
12. Q: Type of expression $7x^2y$7x2y?
    A: Monomial ✅
13. Q: Expand $(x + y)^2$(x+y)2
    A: $x^2 + 2xy + y^2$x2+2xy+y2
14. Q: Expand $(2a − 3b)^2$(2a−3b)2
    A: $4a^2 − 12ab + 9b^2$4a2−12ab+9b2
15. Q: Expand $(x + 2)(x + 5)$(x+2)(x+5)
    A: $x^2 + 7x + 10$x2+7x+10
16. Q: Coefficient of xy in $(x + y)^2$(x+y)2
    A: 2
17. Q: Factorize $x^2 + 2x + 1$x2+2x+1
    A: $(x + 1)^2$(x+1)2
18. Q: Factorize $x^2 − 10x + 25$x2−10x+25
    A: $(x − 5)^2$(x−5)2
19. Q: Expand $(x − 3)(x^2 + 3x + 9)$(x−3)(x2+3x+9)
    A: $x^3 − 27$x3−27
20. Q: True or False: $a^2 − b^2 = (a − b)^2$a2−b2=(a−b)2
    A: False ✅, correct factorization: $(a − b)(a + b)$(a−b)(a+b)