1. Factorize each of the following expressions:
(i) \( 4x^4 + 81y^4 \)
\[(2x^2)^2 + (9y^2)^2 + 2(2x^2)(9y^2) – 2(2x^2)(9y^2)\]
\[(2x^2 + 9y^2)^2 – 36x^2y^2\]
\[(2x^2 + 9y^2)^2 – (6xy)^2\]
\[(2x^2 + 9y^2 + 6xy)(2x^2 + 9y^2 – 6xy)\]
(ii) \( a^4 + 64b^4 \)
\[(a^2)^2 + (8b^2)^2 + 2(a^2)(8b^2) – 2(a^2)(8b^2)\]
\[(a^2 + 8b^2)^2 – 16a^2b^2\]
\[(a^2 + 8b^2)^2 – (4ab)^2\]
\[(a^2 + 8b^2 + 4ab)(a^2 + 8b^2 – 4ab)\]
(iii) \( x^4 + 4x^2 + 16 \)
\[(x^2)^2 + (4)^2 + 2(x^2)(4) – 2(x^2)(4) + 4x^2\]
\[(x^2 + 4)^2 – 8x^2 + 4x^2\]
\[(x^2 + 4)^2 – 4x^2\]
\[(x^2 + 4)^2 – (2x)^2\]
\[(x^2 + 4 + 2x)(x^2 + 4 – 2x)\]
(iv) \( x^4 – 14x^2 + 1 \)
\[(x^2)^2 + (1)^2 + 2(x^2)(1) – 2(x^2)(1) – 14x^2\]
\[(x^2 + 1)^2 – 16x^2\]
\[(x^2 + 1)^2 – (4x)^2\]
\[(x^2 + 1 + 4x)(x^2 + 1 – 4x)\]
(v) \( x^4 – 30x^2y^2 + 9y^4 \)
\[(x^2)^2 + (3y^2)^2 + 2(x^2)(3y^2) – 2(x^2)(3y^2) – 30x^2y^2\]
\[(x^2 + 3y^2)^2 – 6x^2y^2 – 30x^2y^2\]
\[(x^2 + 3y^2)^2 – 36x^2y^2\]
\[(x^2 + 3y^2)^2 – (6xy)^2\]
\[(x^2 + 3y^2 + 6xy)(x^2 + 3y^2 – 6xy)\]
(vi) \( x^4 + 11x^2y^2 + y^4 \)
\[(x^2)^2 + (y^2)^2 + 2(x^2)(y^2) – 2(x^2)(y^2) + 11x^2y^2\]
\[(x^2 + y^2)^2 – 2x^2y^2 + 11x^2y^2\]
\[(x^2 + y^2)^2 + 9x^2y^2\]
\[(x^2 + y^2)^2 – (-9x^2y^2)\]
\[(x^2 + y^2)^2 – (3xy)^2\]
\[(x^2 + y^2 + 3xy)(x^2 + y^2 – 3xy)\]
2. Factorize each of the following expressions:
(i) \( (x+1)(x+2)(x+3)(x+4) + 1 \)
\[(x+1)(x+4) = x^2 + 5x + 4, \quad (x+2)(x+3) = x^2 + 5x + 6\]
\[(x^2 + 5x + 4)(x^2 + 5x + 6) + 1\]
Let \( y = x^2 + 5x \):
\[(y+4)(y+6) + 1 = y^2 + 10y + 24 + 1 = y^2 + 10y + 25\]
\[y^2 + 10y + 25 = (y+5)^2\]
\[(x^2 + 5x + 5)^2\]
(ii) \( (x+2)(x-7)(x-4)(x-1) + 17 \)
\[(x+2)(x-7) = x^2 – 5x – 14, \quad (x-4)(x-1) = x^2 – 5x + 4\]
\[(x^2 – 5x – 14)(x^2 – 5x + 4) + 17\]
Let \( y = x^2 – 5x \):
\[(y-14)(y+4) + 17 = y^2 – 10y – 56 + 17 = y^2 – 10y – 39\]
\[y^2 – 10y – 39 = (y-13)(y+3)\]
\[(x^2 – 5x – 13)(x^2 – 5x + 3)\]
(iii) \( (2x^2+7x+3)(2x^2+7x+5) + 1 \)
Let \( y = 2x^2 + 7x \):
\[(y+3)(y+5) + 1 = y^2 + 8y + 15 + 1 = y^2 + 8y + 16\]
\[y^2 + 8y + 16 = (y+4)^2\]
\[(2x^2 + 7x + 4)^2\]
(iv) \( (3x^2+5x+3)(3x^2+5x+5) – 3 \)
Let \( y = 3x^2 + 5x \):
\[(y+3)(y+5) – 3 = y^2 + 8y + 15 – 3 = y^2 + 8y + 12\]
\[y^2 + 8y + 12 = (y+6)(y+2)\]
\[(3x^2 + 5x + 6)(3x^2 + 5x + 2)\]
(v) \( (x+1)(x+2)(x+3)(x+6) – 3x^2 \)
\[(x+1)(x+6) = x^2 + 7x + 6, \quad (x+2)(x+3) = x^2 + 5x + 6\]
Let \( y = x^2 + 6 \):
\[(y+7x)(y+5x) – 3x^2\]
\[y^2 + 12xy + 35x^2 – 3x^2 = y^2 + 12xy + 32x^2\]
(vi) \( (x+1)(x-2)(x-1)(x+2) + 13x^2 \)
\[(x+1)(x-2) = x^2 – x – 2, \quad (x-1)(x+2) = x^2 + x – 2\]
\[(x^2 – x – 2)(x^2 + x – 2) + 13x^2\]
Let \( y = x^2 – 2 \):
\[(y-x)(y+x) + 13x^2\]
\[y^2 – x^2 + 13x^2 = y^2 + 12x^2\]
\[(x^2 – 2)^2 + 12x^2\]
3. Factorize:
(i) \( 8x^3 + 12x^2 + 6x + 1 \):
\[8x^3 + 12x^2 + 6x + 1 = (2x)^3 + 1^3 + 6x(2x + 1)\]
\[(2x + 1)(4x^2 – 2x + 1) + 6x(2x + 1)\]
\[(2x + 1)(4x^2 – 2x + 6x + 1)\]
\[(2x + 1)(4x^2 + 4x + 1)\]
(ii) \( 27a^3 + 108a^2b + 144ab^2 + 64b^3 \):
\[27a^3 + 64b^3 + 108a^2b + 144ab^2 = (3a)^3 + (4b)^3 + 12ab(3a + 4b)\]
\[(3a + 4b)(9a^2 – 12ab + 16b^2) + 12ab(3a + 4b)\]
\[(3a + 4b)(9a^2 – 12ab + 16b^2 + 12ab)\]
\[(3a + 4b)(9a^2 + 16b^2)\]
(iii) \( x^3 + 48x^2y + 108xy^2 + 216y^3 \):
\[x^3 + 216y^3 + 48x^2y + 108xy^2 = (x)^3 + (6y)^3 + 6xy(x + 6y)\]
\[(x + 6y)(x^2 – 6xy + 36y^2) + 6xy(x + 6y)\]
\[(x + 6y)(x^2 – 6xy + 36y^2 + 6xy)\]
\[(x + 6y)(x^2 + 36y^2)\]
(iv) \( 8x^3 – 125y^3 + 150x^2y – 60xy^2 \):
\[8x^3 – 125y^3 + 150x^2y – 60xy^2 = (2x)^3 – (5y)^3 + 30xy(2x – 5y)\]
\[(2x – 5y)(4x^2 + 10xy + 25y^2) + 30xy(2x – 5y)\]
\[(2x – 5y)(4x^2 + 10xy + 25y^2 + 30xy)\]
\[(2x – 5y)(4x^2 + 40xy + 25y^2)\]
4. Factorize:
(i) \( 125a^3 – 1 \):
\[125a^3 – 1 = (5a)^3 – (1)^3\]
\[(5a – 1)(25a^2 + 5a + 1)\]
(ii) \( 64x^3 + 125 \):
\[64x^3 + 125 = (4x)^3 + (5)^3\]
\[(4x + 5)(16x^2 – 20x + 25)\]
(iii) \( x^6 – 27 \):
\[x^6 – 27 = (x^2)^3 – (3)^3\]
\[(x^2 – 3)(x^4 + 3x^2 + 9)\]
(iv) \( 1000a^3 + 1 \):
\[1000a^3 + 1 = (10a)^3 + (1)^3\]
\[(10a + 1)(100a^2 – 10a + 1)\]
(v) \( 343x^3 + 216 \):
\[343x^3 + 216 = (7x)^3 + (6)^3\]
\[(7x + 6)(49x^2 – 42x + 36)\]
(vi) \( 27 – 512y^3 \):
\[27 – 512y^3 = (3)^3 – (8y)^3\]
\[(3 – 8y)(9 + 24y + 64y^2)\]
