RD Sharma Solutions for Class 8 Math Chapter 7 - Factorization (Part-2) | RD Sharma Solutions for Class 8 Mathematics PDF Download

Question 1: Factorize the following:
3x − 9
Answer 1: The greatest common factor of the terms 3x and -9 of the expression 3x - 9 is 3. 
Now.
3x = 3x
and
-9 = 3.-3
Hence, the expression 3x - 9 can be factorised as 3(x - 3). 
Question 2: Factorize the following:
5x − 15x²

Answer 2: The greatest common factor of the terms 5x and 15x² of the expression 5x - 15x² is 5x. 
Now,
5x = 5x ×× 1 
and
-15x² = 5x ×× -3x
Hence, the expression 5x - 15x² can be factorised as 5x(1 - 3x). 
Question 3: Factorize the following:
20a¹²b² − 15a⁸b⁴

Answer 3: The greatest common factor of the terms 20a¹²b² and -15a⁸b⁴ of the expression 20a¹²b² - 15a⁸b⁴ is 5a⁸b².
20a¹²b² = 5×4×a⁸×a⁴×b² = 5a⁸×b²××4a⁴ and -15a⁸b⁴ = 5×-3×a⁸×b²×b² = 5a⁸b² ×× -3b²
Hence, the expression 20a¹²b² - 15a⁸b⁴ can be factorised as 5a⁸b²(4a⁴-3b²) 

Question 4: Factorize the following:
72x⁶y⁷ − 96x⁷y⁶

Answer 4: The greatest common factor of the terms 72x⁶y⁷ and -96x⁷y⁶ of the expression 72x⁶y⁷ - 96x⁷y⁶⁴ is 24x⁶y⁶.
Now,
72x⁶y⁷ = 24x⁶y⁶ ×× 3y
and 
-96x⁷y⁶ = 24x⁶y⁶ ×× -4x
Hence, the expression 72x⁶y⁷ - 96x⁷y⁶ can be factorised as 24x⁶y⁶(3y - 4x). 

Question 5: Factorize the following:
20x³ − 40x² + 80x

Answer 5: The greatest common factor of the terms 20x³, -40x² and 80x of the expression 20x³ - 40x² + 80x is 20x.
Now,
20x³ = 20x ×× x²
-40x² = 20x ×× -2x
and
80x = 20x ×× 4
Hence, the expression 20x³ - 40x² + 80x can be factorised as 20x(x² - 2x + 4). 
Question 6: Factorize the following:
2x³y² − 4x²y³ + 8xy⁴

Answer 6: The greatest common factor of the terms 2x³y², -4x²y³ and 8xy⁴ of the expression 2x³y² - 4x²y³+ 8xy⁴y⁶⁴ is 2xy². 
Now,
2x³y² = 2xy²  ×× x²  
-4x²y³ = 2xy² ×× -2xy
8xy⁴ = 2xy² ×× 4y²
Hence, the expression 2x³y² - 4x²y³ + 8xy⁴ can be factorised as 2xy²(x² - 2xy + 4y²). 

Question 7: Factorize the following:
10m³n² + 15m⁴n − 20m²n³

Answer 7: The greatest common factor of the terms 10m³n², 15m⁴n and -20m²n³ of the expression 10m³n² + 15m⁴n - 20m²n³ is 5m²n.
Now,
10m³n² = 5m²n ×× 2mn
15m⁴n = 5m²n ×× 3m²
-20m²n³ = 5m²n ×× -4n²
Hence, 10m³n² + 15m²n - 20m²n³ can be factorised as 5m²n(2mn + 3m² - 4n²). 

Question 8: Factorize the following:
2a⁴b⁴ − 3a³b⁵ + 4a²b⁵

Answer 8: The greatest common factor of the terms 2a⁴b⁴, -3a³b⁵ and 4a²b⁵ of the expression 2a⁴b⁴ - 3a³b⁵ + 4a²b⁵ is a²b⁴.
Now,
2a⁴b⁴ = a²b⁴ ×× 2a²
-3a³b⁵ = a²b⁴ ×× -3ab 
4a²b⁵ = a²b⁴ ×× 4b
Hence, (2a⁴b⁴ - 3a³b⁵ + 4a²b⁵) can be factorised as [a²b⁴(2a² - 3ab + 4b)]. 

Question 9: Factorize the following:
28a² + 14a²b² − 21a⁴

Answer 9: The greatest common factor of the terms 28a2, 14a2b2 and 21
a4 of the expression 28a2+14a2b2−21a4 is 7a2. 
Also, we can write 28a2=7a2×4, 14a2b2=7a2×2b2 and 21a4=7a2×3a2. 
∴ 28a2+14a2b2−21a4=7a2×4+7a2×2b2−7a2×3a2 
= 7a2(4+2b2−3a2) 
Question 10: Factorize the following:
a⁴b − 3a²b² − 6ab³ 
Answer 10: The greatest common factor of the terms a4b, 3a2b2 and 6ab3 of the expression a4b−3a2b2−6ab3 is ab. 
Also, we can write a4b=ab×a3, 3a2b2=ab×3ab and 6ab3=ab×6b2. 
∴ a4b−3a2b2−6ab3 = ab×a3−ab×3ab−ab×6b2 
= ab(a3−3ab−6b2) 
Question 11: Factorize the following:
2l²mn - 3lm²n + 4lmn² 
Answer 11: The greatest common factor of the terms 2l2mn, 3lm2n and 4lmn2 of the expression 2l2mn−3lm2n+4lmn2 is lmn. 
Also, we can write 2l2mn=lmn×2l, 3lm2n=lmn×3m and 4lmn2=lmn×4n. 
∴ 2l2nm−3lm2n+4lmn2=lmn×2l−lmn×3m+lmn×4n 
=lmn(2l−3m+4n) 
Question 12: Factorize the following: 

x⁴y² − x²y⁴ − x⁴y⁴ 

Answer 12: The greatest common factor of the terms a4b, 3a2b2 and 6ab3 of the expression a4b−3a2b2−6ab3 is ab. 
Also, we can write a4b=ab×a3, 3a2b2=ab×3ab and 6ab3=ab×6b2. 
∴ a4b−3a2b2−6ab3 = ab×a3−ab×3ab−ab×6b2 
= ab(a3−3ab−6b2) 
Question 11: Factorize the following:
2l²mn - 3lm²n + 4lmn² 
Answer 11: The greatest common factor of the terms 2l2mn, 3lm2n and 4lmn2 of the expression 2l2mn−3lm2n+4lmn2 is lmn. 
Also, we can write 2l2mn=lmn×2l, 3lm2n=lmn×3m and 4lmn2=lmn×4n. 
∴ 2l2nm−3lm2n+4lmn2=lmn×2l−lmn×3m+lmn×4n 
=lmn(2l−3m+4n) 
Question 12: Factorize the following: 

x⁴y² − x²y⁴ − x⁴y⁴ 

Answer 12: The greatest common factor of the terms x4y2, x2y4 and x4y4 of the expression x4y2−x2y4−x4y4 is x2y2. 
Also, we can write x4y2=x2y2×x2, x2y4=x2y2×y2 and x4y4=x2y2×x2y2. ∴ x4y2−x2y4−x4y4 = x2y2×x2−x2y2×y2−x2y2×x2y2 
=x2y2(x2−y2−x2y2) 
Question 13: Factorize the following:
9x²y + 3axy 
Answer 13: The greatest common factor of the terms 9x2y and 3axy of the expression 9x2y+3axy is 3xy. 
Also, we can write 9x2y=3xy×3x and 3axy=3xy×a. 
∴ 9x2y+3axy =3xy×3x+3xy×a 
=3xy(3x+a) 
Question 14: Factorize the following:
16m − 4m²
Answer 14: The greatest common factor of the terms 16m and 4m2 of the expression 16m−4m2 is 4m. 
Also, we can write 16m=4m×4 and 4m2=4m×m.
∴ 16m−4m2=4m×4−4m×m 
=4m(4−m) 
Question 15: Factorize the following:

−4a² + 4ab − 4ca 

Answer 15: The greatest common factor of the terms −4a2, 4ab and− 4ca of the expression−4a2+4ab−4ca is −4a. 
Also, we can write −4a2=−4a×a, 4ab=−4a×(−b) and 4ca=−4a×c. 
∴ −4a2+4ab−4ca=−4a×a+(−4a)×(−b)−4a×c 
=−4a(a−b+c) 
Question 16: Factorize the following:

x²yz + xy²z + xyz² 

Answer 16: The greatest common factor of the terms x2yz, xy2z and xyz2 of the expression x2yz+xy2z+xyz2 is xyz. 
Also, we can write x2yz=xyz×x, xy2z=xyz×y and xyz2=xyz×z. 
∴ x2yz+xy2z+xyz2=xyz×x+xyz×y+xyz×z 
=xyz(x+y+z) 
Question 17:  Factorize the following: 
ax²y + bxy² + cxyz 
Answer 17: The greatest common factor of the terms ax2y, bxy2 and cxyz of the expression ax2y+bxy2+cxyz is xy. 
Also, we can write ax2y=xy×ax, bxy2=xy×by and cxyz=xy×cz.
∴ ax2y+bxy2+cxyz = xy×ax+xy×by+xy×cz  
=xy(ax+by+cz)