Class 8 Maths Factorization Notes & MCQs

Factorization – Quick Notes Class 8

1. Definition:

Factorization is the process of expressing an algebraic expression as a product of its factors.

• Example: $x^2 + 5x = x(x + 5)$x2+5x=x(x+5)

2. Methods of Factorization

1. Common Factor Method
   • Take out the greatest common factor (GCF).
   • Example: $6x^2 + 9x = 3x(2x + 3)$6×2+9x=3x(2x+3)
2. Factorization by Grouping
   • Group terms to factor common elements.
   • Example: $ax + ay + bx + by = (a + b)(x + y)$ax+ay+bx+by=(a+b)(x+y)
3. Factorization of Quadratics
   • $x^2 + (a+b)x + ab = (x + a)(x + b)$x2+(a+b)x+ab=(x+a)(x+b)
   • $x^2 − (a+b)x + ab = (x − a)(x − b)$x2−(a+b)x+ab=(x−a)(x−b)
4. Factorization Using Identities
   • $a^2 − b^2 = (a − b)(a + b)$a2−b2=(a−b)(a+b)
   • $a^2 + 2ab + b^2 = (a + b)^2$a2+2ab+b2=(a+b)2
   • $a^3 + b^3 = (a + b)(a^2 − ab + b^2)$a3+b3=(a+b)(a2−ab+b2)
   • $a^3 − b^3 = (a − b)(a^2 + ab + b^2)$a3−b3=(a−b)(a2+ab+b2)

Tips / Tricks

• Always check GCF first before applying identities.
• Identify perfect square or cube forms.
• For quadratic trinomials, find two numbers whose product = constant term and sum = middle coefficient.

Factorization – MCQ Q&A

1. Q: Factorize $x^2 + 5x$x2+5x
   A: $x(x + 5)$x(x+5)
2. Q: Factorize $6x^2 + 9x$6×2+9x
   A: $3x(2x + 3)$3x(2x+3)
3. Q: Factorize $ax + ay + bx + by$ax+ay+bx+by
   A: $(a + b)(x + y)$(a+b)(x+y)
4. Q: Factorize $x^2 + 7x + 12$x2+7x+12
   A: $(x + 3)(x + 4)$(x+3)(x+4)
5. Q: Factorize $x^2 − 5x + 6$x2−5x+6
   A: $(x − 2)(x − 3)$(x−2)(x−3)
6. Q: Factorize $a^2 − b^2$a2−b2
   A: $(a − b)(a + b)$(a−b)(a+b)
7. Q: Factorize $x^2 + 2xy + y^2$x2+2xy+y2
   A: $(x + y)^2$(x+y)2
8. Q: Factorize $x^3 + y^3$x3+y3
   A: $(x + y)(x^2 − xy + y^2)$(x+y)(x2−xy+y2)
9. Q: Factorize $x^3 − y^3$x3−y3
   A: $(x − y)(x^2 + xy + y^2)$(x−y)(x2+xy+y2)
10. Q: Factorize $2x^2 + 8x$2×2+8x
    A: $2x(x + 4)$2x(x+4)
11. Q: Factorize $x^2 + 9x + 20$x2+9x+20
    A: $(x + 4)(x + 5)$(x+4)(x+5)
12. Q: Factorize $x^2 − 16$x2−16
    A: $(x − 4)(x + 4)$(x−4)(x+4)
13. Q: Factorize $x^2 − 2x − 15$x2−2x−15
    A: $(x − 5)(x + 3)$(x−5)(x+3)
14. Q: Factorize $3x^2 − 12$3×2−12
    A: $3(x^2 − 4) = 3(x − 2)(x + 2)$3(x2−4)=3(x−2)(x+2)
15. Q: Factorize $4x^2 + 12x + 9$4×2+12x+9
    A: $(2x + 3)^2$(2x+3)2
16. Q: Factorize $x^3 + 8$x3+8
    A: $(x + 2)(x^2 − 2x + 4)$(x+2)(x2−2x+4)
17. Q: Factorize $27x^3 − 8$27×3−8
    A: $(3x − 2)(9x^2 + 6x + 4)$(3x−2)(9×2+6x+4)
18. Q: Factorize $5x^2y + 10xy^2$5x2y+10xy2
    A: $5xy(x + 2y)$5xy(x+2y)
19. Q: Factorize $x^2 + 6x + 9$x2+6x+9
    A: $(x + 3)^2$(x+3)2
20. Q: Factorize $x^2 − 10x + 25$x2−10x+25
    A: $(x − 5)^2$(x−5)2