Mathematics (SNC) – Class 9th

Q1. Choose the correct option.

1. $3a^{3}b-9a{b}^2+15ab$3a3b−9ab2+15ab factored is:
   (A) $3ab(a^2-3b+5)$3ab(a2−3b+5)
2. Factorization of $x^2+x-12$x2+x−12:
   (B) $(x-3)(x+4)$(x−3)(x+4)
3. $(x+9)(x+4)$(x+9)(x+4) is the factorized form of:
   (C) $x^2+13x+36$x2+13x+36
4. Factorization of $x^4+4x^2+16$x4+4x2+16:
   (C) $(x^2-2x+4)(x^2+2x+4)$(x2−2x+4)(x2+2x+4)
5. $(x+1)(x+2)(x+3)(x+4)+1=$(x+1)(x+2)(x+3)(x+4)+1=
   (B) $(x^2+5x+5{)}^2$(x2+5x+5)2
6. Factorization of $125a^3-1$125a3−1:
   (C) $(5a-1)(25a^2+5a+1)$(5a−1)(25a2+5a+1)
7. HCF of $a^3+2a^2-3a$a3+2a2−3a and $2a^2+5a-3$2a2+5a−3:
   (A) $a-1$a−1
8. Product of LCM and HCF equals:
   (C) Product
9. To find HCF for $x^2+x$x2+x and $x^3+x^2$x3+x2 with LCM $x^2(x+1)$x2(x+1), we use:
   (B) $\frac{(x^2+x)(x^3+x^2)}{x^2(x+1)}$x2(x+1)(x2+x)(x3+x2)​
10. Factorization of $12x+36$12x+36:
    (A) $12(x+3)$12(x+3)

Q2. Short Answers

[i] $(x+1)(x+3)$(x+1)(x+3)
[ii] $(x-3)(x+4)$(x−3)(x+4)
[iii] $(x-8)(x+7)$(x−8)(x+7)
[iv] $(2y-1)(y-2)$(2y−1)(y−2)
[v] $(x-3)(x^2+3x+9)$(x−3)(x2+3x+9)
[vi] LCM: $8x^2(x+3)$8x2(x+3), HCF: $4x$4x
[vii] LCM: $x(x-3)(x+2)$x(x−3)(x+2), HCF: $x-3$x−3
[viii] Algebraic factorization: Expressing an expression as a product of factors. Example: $x^2-4=(x-2)(x+2)$x2−4=(x−2)(x+2)
[ix] $(x-3)(x-8)$(x−3)(x−8)
[x] $(2x+5)(x+3)$(2x+5)(x+3)
[xi] $(x^2-3y^2{)}^2$(x2−3y2)2
[xii] $(2x-5y)(4x^2+10xy+25y^2)$(2x−5y)(4x2+10xy+25y2)
[xiii] LCM: $8x^2{y}^2$8x2y2
[xiv] HCF: $a(a-1)$a(a−1)
[xv] Other polynomial: $y^2-5y-14$y2−5y−14

Q3. Detailed Answers (Answer any 2)

1. Factorize: a4+64b4a4+64b4

$$a^4+64b^4=(a^2{)}^2+(8b^2{)}^2$$a4+64b4=(a2)2+(8b2)2

Add and subtract $16a^2{b}^2$16a2b2:$$=a^4+16a^2{b}^2+64b^4-16a^2{b}^2$$=a4+16a2b2+64b4−16a2b2$$=(a^2+8b^2{)}^2-(4ab{)}^2$$=(a2+8b2)2−(4ab)2

Using difference of squares:$$=(a^2+8b^2-4ab)(a^2+8b^2+4ab)$$=(a2+8b2−4ab)(a2+8b2+4ab)

✅ Factors: $(a^2-4ab+8b^2)(a^2+4ab+8b^2)$(a2−4ab+8b2)(a2+4ab+8b2)

2. Factorize: (x+1)(x−1)(x+2)(x−2)−16×2(x+1)(x−1)(x+2)(x−2)−16x2

Group:$$[(x+1)(x-1)][(x+2)(x-2)]-16x^2$$[(x+1)(x−1)][(x+2)(x−2)]−16x2$$(x^2-1)(x^2-4)-16x^2$$(x2−1)(x2−4)−16x2

Let $y=x^2$y=x2:$$(y-1)(y-4)-16y=y^2-5y+4-16y$$(y−1)(y−4)−16y=y2−5y+4−16y$$=y^2-21y+4$$=y2−21y+4

Factor:$$=(y-\frac{21+\sqrt{425}}{2})(y-\frac{21-\sqrt{425}}{2})$$=(y−221+425​​)(y−221−425​​)

Substitute back $y=x^2$y=x2:
✅ Factors: $(x^2-\frac{21+5\sqrt{17}}{2})(x^2-\frac{21-5\sqrt{17}}{2})$(x2−221+517​​)(x2−221−517​​)

3. Find product of two polynomials

Given:
HCF = $x+a$x+a
LCM = $12x^2(x+a)(x-a^2)$12x2(x+a)(x−a2)

We know:$$\text{Product of polynomials}=\text{HCF}\times \text{LCM}$$Product of polynomials=HCF×LCM$$=(x+a)\times 12x^2(x+a)(x-a^2)$$=(x+a)×12x2(x+a)(x−a2)$$=12x^2(x+a{)}^2(x-a^2)$$=12x2(x+a)2(x−a2)

✅ Product: $12x^2(x+a{)}^2(x-a^2)$12x2(x+a)2(x−a2)

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