Find the common factors of the given terms.

1. 12x, 36         

2. 2y, 22xy        

3. 14pq, 28 p² q²     

4. 2x, 3x², 4      

5. 6abc, 24ab², 12a² b            

6. 16x³, -4x³, 32x      

7. 10pq, 20qr, 30rp            

8. 3x² y³, 10x³ y³, 6 x² y² z

Factorize the following expressions.

1. 7x – 42       

2. 6p – 12q      

3. 7a² + 14a          

4. -16z + 20z³            

5. 20l² m + 30alm    

6. 5x² y – 15xy²         

7. 10a² – 15b² + 20c²        

8. -4a² + 4ab – 4ca   

9.  x² yz + xy² z + xyz²

10. ax² y + bxy² + cxyz

Factorize:

1. x² + xy + 8x + 8y                         

2. 15xy – 6x + 5y – 2              

3. ax + bx – ay – by

4. 15pq + 15 + 9q + 25p              

5. z – 7 + 7xy – xyz

Factorize the following expressions:

1. a² + 8a + 16        

2. p² – 10p + 25           

3. 25m² + 30m + 9               

4. 49y² + 84yz + 36z²

5. 4x² – 8x + 4       

6. 121b² – 88bc + 16c²     

7. (l + m)² – 4lm            

8. a⁴ + 2a² b² + b⁴

Factorize:

1. 4p² – 9q²        

2. 63a² – 112b²        

3. 9x² – 36      4. 16x⁵ – 144x²      

5. (l + m)² – (l - m)²

6. 9x² y² – 16          

7. (x² – 2xy + y²) – z²            

8. 25a² – 4b² + 28bc – 49c²

Factorize the expressions:

1. ax² + bx            

2. 7p² + 21q²           

3. 2x³ + 2xy² + 2xz²               

4. am² + bm² + bn² + an

5. (lm + l) + m + 1   

6. y (y + z) + 9(y + z)     

7. 5y² – 20y – 8z + 2yz  

8. 10ab + 4a + 5b + 2

9. 6xy – 4y + 6 – 9x

Factorize:

1. a⁴ – b⁴        

2. p⁴ – 81        

3. x⁴ – (y + z)⁴         

4. x⁴ – (x – z)⁴         

5. a⁴ – 2a² b² + b⁴

Factorize the following expressions:

1. p² + 6p + 8                    

2.  q² – 10q + 21                      

3. p² + 6p – 16

Carry out the following divisions:

1. 28x⁴ ÷ 56x                               

2. -36y³ ÷ 9y²                            

3. 66pq² r³ ÷ 11qr² 

4. 34x³ y³ z³ ÷ 51xy² z³           

5. 12a⁸ b⁸ ÷ (-6a⁶ b⁴)

Divide the given polynomial by the given monomial:

1. (5x² – 6x) ÷ 3x      

2. (3y⁸ – 4y⁶ + 5y⁴) ÷ y⁴      

3. 8(x³ y² z² + x² y³ z² + x² y^{2 3}z) ÷4 x² y² z²

4. (x³ + 2x² + 3x) ÷ 2x     

5. (p³ q⁶ – p⁶ q³) ÷ p³ q³

Work out the following divisions:

1. (10x - 25) ÷ 5                       

2. (10x - 25) ÷ (2x - 5)              

3. 10y (6y + 21) ÷ 5(2y + 7)

4. 9x² y²(3z - 24) ÷ 27xy (z - 8)       

5. 96abc (3a - 12) (5b - 30) ÷ 144(a - 4) (a - 6)

Divide as directed:

1. 5(2x + 1) (3x + 5) ÷ (2x + 1)                                              

2. 26xy (x + 5) (y - 4) ÷ 13x (y - 4)

3. 52pqr (p + q) (q + r) (r + p) ÷ 104pq (q + r) (r + p)         

4. 20(y + 4) (y² + 5y + 3) ÷ 5(y + 4)

5. x (x + 1) (x + 2) (x + 3) ÷ x (x + 1)

Factorize the expressions and divide them as directed:

1. (y² + 7y + 10) ÷ (y + 5)

2. (m² – 14m - 32) ÷ (m + 2)

3. (5p² – 25p + 20) ÷ (p - 1)

4. 4yz (z² + 6z - 16) ÷ 2y (z + 8)

5. 5pq (p² – q²) ÷ 2p (p + q)

6. 12xy (9x² – 16y²) ÷ 4xy (3x + 4y)

7. 39y³(50y² - 98) ÷ 26y²(5y + 7)

Find and correct the errors in the following mathematical statement: 4(x – 5) = 4x – 5

Find and correct the errors in the following mathematical statement: x (3x + 2) = 3x² + 2

Find and correct the errors in the following mathematical statement: 2x + 3y = 5xy

Find and correct the errors in the following mathematical statement: x + 2x + 3x = 5x

Find and correct the errors in the following mathematical statement: 5y + 2y + y – 7y = 0

Find and correct the errors in the following mathematical statement: 3x + 2x = 5x²

Find and correct the errors in the following mathematical statement:

(2x)² + 4(2x) + 7 = 2x² + 8x + 7

Find and correct the errors in the following mathematical statement:

(2x)² + 5x = 4x + 5x = 9x

Find and correct the errors in the following mathematical statement:

(3x + 2)² = 3x² + 6x + 9

Find and correct the errors in the following mathematical statements:

Substituting x = -3 in:

(a) x² + 5x + 4 gives (-3)² + 5(-3) + 4 = 9 + 2 + 4 = 15

(b) x² - 5x + 4 gives (-3)² - 5(-3) + 4 = 9 - 15 + 4 = 15

(c) x² + 5x + 4 gives (-3)² + 5(-3) = -9 - 15 = -24

Find and correct the errors in the following mathematical statement: (y - 3)² = y² – 9

Find and correct the errors in the following mathematical statement: (z + 5)² = z² + 25

Find and correct the errors in the following mathematical statement:

(2a + 3b) (a - b) = 2a² – 3b²

Find and correct the errors in the following mathematical statement:

(a + 4) (a + 2) = a² + 8

Find and correct the errors in the following mathematical statement:

(a - 4) (a - 2) = a² – 8

Find and correct the errors in the following mathematical statement: 3x²/3x² = 0

Find and correct the errors in the following mathematical statement: (3x² + 1) ÷ 3x² = 1 + 1 = 2

Find and correct the errors in the following mathematical statement: 3x ÷ (3x + 2) = 1 ÷ 2

Find and correct the errors in the following mathematical statement: 3 ÷ (4x + 3) = 1 ÷ 4x

Find and correct the errors in the following mathematical statement: (4x + 5) ÷ 4x = 5

Find and correct the errors in the following mathematical statement: (7x + 5) ÷ 5 = 7x