Exercise 4.3: Factorization
Factorize:
1. $a^3 + 8$
Solution: $a^3 + 8$
$= a^3 + 2^3$
$= (a + 2)(a^2 – a \cdot 2 + 2^2)$
$= (a + 2)(a^2 – 2a + 4)$ (Answer)
2. $8x^3 + 343$
Solution: $8x^3 + 343$
$= (2x)^3 + 7^3$
$= (2x + 7)\{(2x)^2 – 2x \cdot 7 + 7^2\}$
$= (2x + 7)(4x^2 – 14x + 49)$ (Answer)
3. $8a^4 + 27ab^3$
Solution: $8a^4 + 27ab^3$
$= a(8a^3 + 27b^3)$
$= a\{(2a)^3 + (3b)^3\}$
$= a(2a + 3b)\{(2a)^2 – 2a \cdot 3b + (3b)^2\}$
$= a(2a + 3b)(4a^2 – 6ab + 9b^2)$ (Answer)
4. $8x^3 + 1$
Solution: $8x^3 + 1$
$= (2x)^3 + 1^3$
$= (2x + 1)\{(2x)^2 – 2x \cdot 1 + 1^2\}$
$= (2x + 1)(4x^2 – 2x + 1)$ (Answer)
5. $64a^3 – 125b^3$
Solution: $64a^3 – 125b^3$
$= (4a)^3 – (5b)^3$
$= (4a – 5b)\{(4a)^2 + 4a \cdot 5b + (5b)^2\}$
$= (4a – 5b)(16a^2 + 20ab + 25b^2)$ (Answer)
6. $729a^3 – 64b^3c^6$
Solution: $729a^3 – 64b^3c^6$
$= (9a)^3 – (4bc^2)^3$
$= (9a – 4bc^2)\{(9a)^2 + 9a \cdot 4bc^2 + (4bc^2)^2\}$
$= (9a – 4bc^2)(81a^2 + 36abc^2 + 16b^2c^4)$ (Answer)
7. $27a^3b^3 + 64b^3c^3$
Solution: $27a^3b^3 + 64b^3c^3$
$= b^3(27a^3 + 64c^3)$
$= b^3\{(3a)^3 + (4c)^3\}$
$= b^3(3a + 4c)\{(3a)^2 – 3a \cdot 4c + (4c)^2\}$
$= b^3(3a + 4c)(9a^2 – 12ac + 16c^2)$ (Answer)
8. $56x^3 – 189y^3$
Solution: $56x^3 – 189y^3$
$= 7(8x^3 – 27y^3)$
$= 7\{(2x)^3 – (3y)^3\}$
$= 7(2x – 3y)\{(2x)^2 + 2x \cdot 3y + (3y)^2\}$
$= 7(2x – 3y)(4x^2 + 6xy + 9y^2)$ (Answer)
9. $3x – 75x^3$
Solution: $3x – 75x^3$
$= 3x(1 – 25x^2)$
$= 3x\{1^2 – (5x)^2\}$
$= 3x(1 + 5x)(1 – 5x)$ (Answer)
10. $4x^2 – y^2$
Solution: $4x^2 – y^2$
$= (2x)^2 – y^2$
$= (2x + y)(2x – y)$ (Answer)
11. $3ay^2 – 48a$
Solution: $3ay^2 – 48a$
$= 3a(y^2 – 16)$
$= 3a(y^2 – 4^2)$
$= 3a(y + 4)(y – 4)$ (Answer)
12. $a^2 – 2ab + b^2 – p^2$
Solution: $a^2 – 2ab + b^2 – p^2$
$= (a – b)^2 – p^2$
$= (a – b + p)(a – b – p)$ (Answer)
13. $16y^2 – a^2 – 6a – 9$
Solution: $16y^2 – a^2 – 6a – 9$
$= (4y)^2 – (a^2 + 6a + 9)$
$= (4y)^2 – (a^2 + 2 \cdot a \cdot 3 + 3^2)$
$= (4y)^2 – (a + 3)^2$
$= \{4y + (a + 3)\}\{4y – (a + 3)\}$
$= (4y + a + 3)(4y – a – 3)$ (Answer)
14. $8a + ap^3$
Solution: $8a + ap^3$
$= a(8 + p^3)$
$= a(2^3 + p^3)$
$= a(2 + p)(2^2 – 2 \cdot p + p^2)$
$= a(2 + p)(4 – 2p + p^2)$ (Answer)
15. $2a^3 + 16b^3$
Solution: $2a^3 + 16b^3$
$= 2(a^3 + 8b^3)$
$= 2\{a^3 + (2b)^3\}$
$= 2(a + 2b)(a^2 – a \cdot 2b + (2b)^2)$
$= 2(a + 2b)(a^2 – 2ab + 4b^2)$ (Answer)
16. $x^2 + y^2 – 2xy – 1$
Solution: $x^2 + y^2 – 2xy – 1$
$= (x^2 – 2xy + y^2) – 1$
$= (x – y)^2 – 1^2$
$= (x – y + 1)(x – y – 1)$ (Answer)
17. $a^2 – 2ab + 2b – 1$
Solution: $a^2 – 2ab + 2b – 1$
$= (a^2 – 1) – (2ab – 2b)$
$= (a + 1)(a – 1) – 2b(a – 1)$
$= (a – 1)(a + 1 – 2b)$ (Answer)
18. $x^4 – 2x^2 + 1$
Solution: $x^4 – 2x^2 + 1$
$= (x^2)^2 – 2 \cdot x^2 \cdot 1 + 1^2$
$= (x^2 – 1)^2$
$= \{(x + 1)(x – 1)\}^2$
$= (x + 1)^2 (x – 1)^2$ (Answer)
19. $36 – 12x + x^2$
Solution: $36 – 12x + x^2$
$= 6^2 – 2 \cdot 6 \cdot x + x^2$
$= (6 – x)^2$ (Answer)
20. $x^6 – y^6$
Solution: $x^6 – y^6$
$= (x^3)^2 – (y^3)^2$
$= (x^3 + y^3)(x^3 – y^3)$
$= (x + y)(x^2 – xy + y^2)(x – y)(x^2 + xy + y^2)$
$= (x + y)(x – y)(x^2 + xy + y^2)(x^2 – xy + y^2)$ (Answer)
21. $(x – y)^3 + z^3$
Solution: $(x – y)^3 + z^3$
$= \{(x – y) + z\}\{(x – y)^2 – (x – y)z + z^2\}$
$= (x – y + z)(x^2 – 2xy + y^2 – xz + yz + z^2)$ (Answer)
22. $64x^3 – 8y^3$
Solution: $64x^3 – 8y^3$
$= 8(8x^3 – y^3)$
$= 8\{(2x)^3 – y^3\}$
$= 8(2x – y)\{(2x)^2 + 2x \cdot y + y^2\}$
$= 8(2x – y)(4x^2 + 2xy + y^2)$ (Answer)
23. $x^2 + 14x + 40$
Solution: $x^2 + 14x + 40$
$= x^2 + 10x + 4x + 40$
$= x(x + 10) + 4(x + 10)$
$= (x + 10)(x + 4)$ (Answer)
24. $x^2 + 7x – 120$
Solution: $x^2 + 7x – 120$
$= x^2 + 15x – 8x – 120$
$= x(x + 15) – 8(x + 15)$
$= (x + 15)(x – 8)$ (Answer)
25. $x^2 – 51x + 650$
Solution: $x^2 – 51x + 650$
$= x^2 – 26x – 25x + 650$
$= x(x – 26) – 25(x – 26)$
$= (x – 26)(x – 25)$ (Answer)
26. $a^2 + 7ab + 12b^2$
Solution: $a^2 + 7ab + 12b^2$
$= a^2 + 4ab + 3ab + 12b^2$
$= a(a + 4b) + 3b(a + 4b)$
$= (a + 4b)(a + 3b)$ (Answer)
27. $p^2 + 2pq – 80q^2$
Solution: $p^2 + 2pq – 80q^2$
$= p^2 + 10pq – 8pq – 80q^2$
$= p(p + 10q) – 8q(p + 10q)$
$= (p + 10q)(p – 8q)$ (Answer)
28. $x^2 – 3xy – 40y^2$
Solution: $x^2 – 3xy – 40y^2$
$= x^2 – 8xy + 5xy – 40y^2$
$= x(x – 8y) + 5y(x – 8y)$
$= (x – 8y)(x + 5y)$ (Answer)
29. $(x^2 – x)^2 + 3(x^2 – x) – 40$
Solution: Let, $x^2 – x = a$
Given expression $= a^2 + 3a – 40$
$= a^2 + 8a – 5a – 40$
$= a(a + 8) – 5(a + 8)$
$= (a + 8)(a – 5)$
$= (x^2 – x + 8)(x^2 – x – 5)$ [Substituting the value of $a$] (Answer)
30. $(a^2 + b^2)^2 – 18(a^2 + b^2) – 88$
Solution: Let, $a^2 + b^2 = x$
Given expression $= x^2 – 18x – 88$
$= x^2 – 22x + 4x – 88$
$= x(x – 22) + 4(x – 22)$
$= (x – 22)(x + 4)$
$= (a^2 + b^2 – 22)(a^2 + b^2 + 4)$ [Substituting the value of $x$] (Answer)
31. $(a^2 + 7a)^2 – 8(a^2 + 7a) – 180$
Solution: Let, $a^2 + 7a = x$
Given expression $= x^2 – 8x – 180$
$= x^2 – 18x + 10x – 180$
$= x(x – 18) + 10(x – 18)$
$= (x – 18)(x + 10)$
$= (a^2 + 7a – 18)(a^2 + 7a + 10)$ [Substituting the value of $x$]
$= (a^2 + 9a – 2a – 18)(a^2 + 5a + 2a + 10)$
$= \{a(a + 9) – 2(a + 9)\}\{a(a + 5) + 2(a + 5)\}$
$= (a + 9)(a – 2)(a + 5)(a + 2)$ (Answer)
32. $x^2 + (3a + 4b)x + (2a^2 + 5ab + 3b^2)$
Solution: First, let us factorize the expression $2a^2 + 5ab + 3b^2$:
$2a^2 + 5ab + 3b^2$
$= 2a^2 + 2ab + 3ab + 3b^2$
$= 2a(a + b) + 3b(a + b)$
$= (a + b)(2a + 3b)$
Original expression $= x^2 + (3a + 4b)x + (a + b)(2a + 3b)$
$= x^2 + \{(a + b) + (2a + 3b)\}x + (a + b)(2a + 3b)$
$= x^2 + (a + b)x + (2a + 3b)x + (a + b)(2a + 3b)$
$= x\{x + (a + b)\} + (2a + 3b)\{x + (a + b)\}$
$= (x + a + b)(x + 2a + 3b)$ (Answer)
33. $6x^2 – x – 15$
Solution: $6x^2 – x – 15$
$= 6x^2 – 10x + 9x – 15$
$= 2x(3x – 5) + 3(3x – 5)$
$= (3x – 5)(2x + 3)$ (Answer)
34. $x^2 – x – (a + 1)(a + 2)$
Solution: Here, $(a + 2) – (a + 1) = 1$
Given expression $= x^2 – \{(a + 2) – (a + 1)\}x – (a + 1)(a + 2)$
$= x^2 – (a + 2)x + (a + 1)x – (a + 1)(a + 2)$
$= x\{x – (a + 2)\} + (a + 1)\{x – (a + 2)\}$
$= \{x – (a + 2)\}\{x + (a + 1)\}$
$= (x – a – 2)(x + a + 1)$ (Answer)
35. $3x^2 + 11x – 4$
Solution: $3x^2 + 11x – 4$
$= 3x^2 + 12x – x – 4$
$= 3x(x + 4) – 1(x + 4)$
$= (x + 4)(3x – 1)$ (Answer)
36. $3x^2 – 16x – 12$
Solution: $3x^2 – 16x – 12$
$= 3x^2 – 18x + 2x – 12$
$= 3x(x – 6) + 2(x – 6)$
$= (x – 6)(3x + 2)$ (Answer)
37. $2x^2 – 9x – 35$
Solution: $2x^2 – 9x – 35$
$= 2x^2 – 14x + 5x – 35$
$= 2x(x – 7) + 5(x – 7)$
$= (x – 7)(2x + 5)$ (Answer)
38. $2x^2 – 5xy + 2y^2$
Solution: $2x^2 – 5xy + 2y^2$
$= 2x^2 – 4xy – xy + 2y^2$
$= 2x(x – 2y) – y(x – 2y)$
$= (x – 2y)(2x – y)$ (Answer)
39. $x^3 – 8(x – y)^3$
Solution: $x^3 – 8(x – y)^3$
$= x^3 – \{2(x – y)\}^3$
$= x^3 – (2x – 2y)^3$
$= \{x – (2x – 2y)\}\{x^2 + x(2x – 2y) + (2x – 2y)^2\}$
$= (x – 2x + 2y)(x^2 + 2x^2 – 2xy + 4x^2 – 8xy + 4y^2)$
$= (2y – x)(7x^2 – 10xy + 4y^2)$ (Answer)
40. $10p^2 + 11pq – 6q^2$
Solution: $10p^2 + 11pq – 6q^2$
$= 10p^2 + 15pq – 4pq – 6q^2$
$= 5p(2p + 3q) – 2q(2p + 3q)$
$= (2p + 3q)(5p – 2q)$ (Answer)
41. $2(x + y)^2 – 3(x + y) – 2$
Solution: Let, $x + y = a$
Given expression $= 2a^2 – 3a – 2$
$= 2a^2 – 4a + a – 2$
$= 2a(a – 2) + 1(a – 2)$
$= (a – 2)(2a + 1)$
Now substituting the value of $a$, we get:
$= (x + y – 2)\{2(x + y) + 1\}$
$= (x + y – 2)(2x + 2y + 1)$ (Answer)
42. $ax^2 + (a^2 + 1)x + a$
Solution: $ax^2 + (a^2 + 1)x + a$
$= ax^2 + a^2x + x + a$
$= ax(x + a) + 1(x + a)$
$= (x + a)(ax + 1)$ (Answer)
43. $15x^2 – 11xy – 12y^2$
Solution: $15x^2 – 11xy – 12y^2$
$= 15x^2 – 20xy + 9xy – 12y^2$
$= 5x(3x – 4y) + 3y(3x – 4y)$
$= (3x – 4y)(5x + 3y)$ (Answer)
44. $a^3 – 3a^2b + 3ab^2 – 2b^3$
Solution: $a^3 – 3a^2b + 3ab^2 – 2b^3$
$= a^3 – 3a^2b + 3ab^2 – b^3 – b^3$
$= (a – b)^3 – b^3$
$= \{(a – b) – b\}\{(a – b)^2 + (a – b)b + b^2\}$
$= (a – 2b)(a^2 – 2ab + b^2 + ab – b^2 + b^2)$
$= (a – 2b)(a^2 – ab + b^2)$ (Answer)