Factorization

Q1: x² − 81

Step 1: Recognize the form:
x² − 81 = x² − 9²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Here, A = x and B = 9
Step 3: Write the factors:
= (x − 9)(x + 9)
Answer: (x − 9)(x + 9)

Q2: 9a² − 25

Step 1: Recognize the form:
9a² − 25 = (3a)² − 5²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Here, A = 3a and B = 5
Step 3: Write the factors:
= (3a − 5)(3a + 5)
Answer: (3a − 5)(3a + 5)

Q3: 36y² − 121

Step 1: Recognize the form:
36y² − 121 = (6y)² − 11²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Here, A = 6y and B = 11
Step 3: Write the factors:
= (6y − 11)(6y + 11)
Answer: (6y − 11)(6y + 11)

Q4: 49a² − 100b²

Step 1: Recognize the form:
49a² − 100b² = (7a)² − (10b)²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Here, A = 7a and B = 10b
Step 3: Write the factors:
= (7a − 10b)(7a + 10b)
Answer: (7a − 10b)(7a + 10b)

Q5: (a+b)² − 36

Step 1: Recognize the form:
(a+b)² − 36 = (a+b)² − (6)²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Here, A = (a+b) and B = 6
Step 3: Write the factors:
= (a+b − 6)(a+b + 6)
Answer: (a+b − 6)(a+b + 6)

Q6: 16c² − 1

Step 1: Recognize the form:
16c² − 1 = (4c)² − 1²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Here, A = 4c and B = 1
Step 3: Write the factors:
= (4c − 1)(4c + 1)
Answer: (4c − 1)(4c + 1)

Q7: 1 − 64b²

Step 1: Recognize the form:
1 − 64b² = 1² − (8b)²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Here, A = 1 and B = 8b
Step 3: Write the factors:
= (1 − 8b)(1 + 8b)
Answer: (1 − 8b)(1 + 8b)

Q8: \(\frac{9}{16} – 25x^2\)

Step 1: Recognize that this is a difference of squares: \[ \frac{9}{16} = \left(\frac{3}{4}\right)^2,\quad 25x^2 = \left(5x\right)^2 \]Step 2: Apply the identity \(a^2 – b^2 = (a – b)(a + b)\): \[ = \left(\frac{3}{4} – 5x\right)\left(\frac{3}{4} + 5x\right) \]Step 3: Write the final factored form: \[ = \left(\frac{3 – 20x}{4}\right)\left(\frac{3 + 20x}{4}\right) \] Or you can leave it as: \[ = \left(\frac{3}{4} – 5x\right)\left(\frac{3}{4} + 5x\right) \]Answer: \(\left(\frac{3}{4} – 5x\right)\left(\frac{3}{4} + 5x\right)\)

Q9: \(z^2 – \frac{1}{144}\)

Step 1: Recognize that this is a difference of squares: \[ z^2 = (z)^2,\quad \frac{1}{144} = \left(\frac{1}{12}\right)^2 \]Step 2: Apply the identity \(a^2 – b^2 = (a – b)(a + b)\): \[ = \left(z – \frac{1}{12}\right)\left(z + \frac{1}{12}\right) \]Step 3: Write the final factored form: \[ = \left(z – \frac{1}{12}\right)\left(z + \frac{1}{12}\right) \]Answer: \(\left(z – \frac{1}{12}\right)\left(z + \frac{1}{12}\right)\)

Q10: 1 − (a−b)²

Step 1: Recognize the form:
1 − (a−b)² = 1² − (a−b)²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Here, A = 1 and B = (a−b)
Step 3: Write the factors:
= (1 − (a−b))(1 + (a−b))
= (1 − a + b)(1 + a − b)
Answer: 1 − (a−b)² = (1 + a − b)(1 + b − a)

Q11: (3m−n)² − (m−2n)²

Step 1: Recognize the form:
(3m−n)² − (m−2n)² = A² − B²
Here, A = (3m−n), B = (m−2n)
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Step 3: Write the factors:
= ((3m−n) − (m−2n))((3m−n) + (m−2n))
= (2m + n)(4m − 3n)
Answer: (3m−n)² − (m−2n)² = (2m + n)(4m − 3n)

Q12: (3x + 2y)² − (2x − 3y)²

Step 1: Recognize the form:
(3x + 2y)² − (2x − 3y)² = A² − B²
Here, A = (3x + 2y), B = (2x − 3y)
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Step 3: Write the factors:
= ((3x + 2y) − (2x − 3y))((3x + 2y) + (2x − 3y))
= (x + 5y)(5x − y)
Answer: (3x + 2y)² − (2x − 3y)² = (x + 5y)(5x − y)

Q13: 16(a+b)² − 9(a−b)²

Step 1: Recognize the form:
16(a+b)² − 9(a−b)² = (4(a+b))² − (3(a−b))²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Step 3: Write the factors:
= (4(a+b) − 3(a−b))(4(a+b) + 3(a−b))
= (a + 7b)(7a + b)
Answer: (a + 7b)(7a + b)

Q14: 9(x+y)² − 16(x−2y)²

Step 1: Recognize the form:
9(x+y)² − 16(x−2y)² = (3(x+y))² − (4(x−2y))²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Step 3: Write the factors:
= (3(x+y) − 4(x−2y))(3(x+y) + 4(x−2y))
= (11y − x)(7x − 5y)
Answer: (11y − x)(7x − 5y)

Q15: 36(a-b)² − 25(a+b)²

Step 1: Recognize the form:
36(a-b)² − 25(a+b)² = (6(a-b))² − (5(a+b))²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Step 3: Write the factors:
= (6(a-b) − 5(a+b))(6(a-b) + 5(a+b))
= (a − 11b)(11a − b)
Answer: (a − 11b)(11a − b)

Q16: 9(3x+1)² − 4(x−1)²

Step 1: Recognize the form:
9(3x+1)² − 4(x−1)² = (3(3x+1))² − (2(x−1))²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Step 3: Write the factors:
= (3(3x+1) − 2(x−1))(3(3x+1) + 2(x−1))
= (7x + 5)(11x + 1)
Answer: (7x + 5)(11x + 1)

Q17: a² − 2ab + b² − c²

Step 1: Recognize the forms:
a² − 2ab + b² − c² = (a-b)² − c²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Step 3: Write the factors:
= ((a-b) − c)((a-b) + c)
= (a − b − c)(a − b + c)
Answer: (a − b − c)(a − b + c)

Q18: x² − a² − 2a − 1

Step 1: Group terms:
x² − a² − 2a − 1 = x² − (a² + 2a + 1) = x² − (a+1)²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Step 3: Write the factors:
= (x − (a+1))(x + (a+1))
= (x − a − 1)(x + a + 1)
Answer: (x − a − 1)(x + a + 1)

Q19: x² − m² + 6mn − 9n²

Step 1: Group terms:
x² − m² + 6mn − 9n² = x² − (m² − 6mn + 9n²) = x² − (m − 3n)²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Step 3: Write the factors:
= (x − (m − 3n))(x + (m − 3n))
= (x − m + 3n)(x + m − 3n)
Answer: (x − m + 3n)(x + m − 3n)

Q20: a⁴ − b⁴

Step 1: Recognize the form:
a⁴ − b⁴ = (a²)² − (b²)²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Step 3: Factor further:
a² − b² = (a − b)(a + b)
So, a⁴ − b⁴ = (a − b)(a + b)(a² + b²)
Answer: (a − b)(a + b)(a² + b²)

Q21: 16a⁴ − 81b⁴

Step 1: Recognize the form:
16a⁴ − 81b⁴ = (4a²)² − (9b²)²
Step 2: Apply the identity A² − B² = (A − B)(A + B).
Step 3: Factor further:
4a² − 9b² = (2a − 3b)(2a + 3b)
So, 16a⁴ − 81b⁴ = (2a − 3b)(2a + 3b)(4a² + 9b²)
Answer: (2a − 3b)(2a + 3b)(4a² + 9b²)

Q22: 3 − 75z²

Step 1: Factor out the common factor:
3 − 75z² = 3(1 − 25z²)
Step 2: Recognize the difference of squares:
1 − 25z² = 1² − (5z)²
Step 3: Apply the identity A² − B² = (A − B)(A + B):
1² − (5z)² = (1 − 5z)(1 + 5z)
Step 4: Write the complete factorisation:
3 − 75z² = 3(1 − 5z)(1 + 5z)
Answer: 3(1 − 5z)(1 + 5z)

Q23: 48a²b² − 3

Step 1: Factor out the common factor:
48a²b² − 3 = 3(16a²b² − 1)
Step 2: Recognize the difference of squares:
16a²b² − 1 = (4ab)² − 1²
Step 3: Apply the identity A² − B² = (A − B)(A + B):
(4ab)² − 1² = (4ab − 1)(4ab + 1)
Step 4: Write the complete factorisation:
48a²b² − 3 = 3(4ab − 1)(4ab + 1)
Answer: 3(4ab − 1)(4ab + 1)

Q24: 4x³ − 81x

Step 1: Factor out the common factor:
4x³ − 81x = x(4x² − 81)
Step 2: Recognize the difference of squares:
4x² − 81 = (2x)² − 9²
Step 3: Apply the identity A² − B² = (A − B)(A + B):
(2x)² − 9² = (2x − 9)(2x + 9)
Step 4: Write the complete factorisation:
4x³ − 81x = x(2x − 9)(2x + 9)
Answer: x(2x − 9)(2x + 9)

Q25: 9b³ − 144b

Step 1: Factor out the common factor:
9b³ − 144b = 9b(b² − 16)
Step 2: Recognize the difference of squares:
b² − 16 = b² − 4²
Step 3: Apply the identity A² − B² = (A − B)(A + B):
b² − 4² = (b − 4)(b + 4)
Step 4: Write the complete factorisation:
9b³ − 144b = 9b(b − 4)(b + 4)
Answer: 9b(b − 4)(b + 4)

Q26: \(32x^2 – 72y^2\)

Step 1: Find the greatest common factor (GCF) of the terms: \[ GCF = 8 \\ 32x^2 – 72y^2 = 8(4x^2 – 9y^2) \]Step 2: Recognize that \(4x^2 – 9y^2\) is a difference of squares: \[ 4x^2 = (2x)^2,\quad 9y^2 = (3y)^2 \]Step 3: Apply the identity \(a^2 – b^2 = (a – b)(a + b)\): \[ = 8(2x – 3y)(2x + 3y) \]Answer: 8(2x – 3y)(2x + 3y)

Q27: \(50x^2y – 32y^3\)

Step 1: Find the greatest common factor (GCF) of the terms: \[ GCF = 2y \\ 50x^2y – 32y^3 = 2y(25x^2 – 16y^2) \]Step 2: Recognize that \(25x^2 – 16y^2\) is a difference of squares: \[ 25x^2 = (5x)^2,\quad 16y^2 = (4y)^2 \]Step 3: Apply the identity \(a^2 – b^2 = (a – b)(a + b)\): \[ = 2y(5x – 4y)(5x + 4y) \]Answer: 2y(5x – 4y)(5x + 4y)

Q28: a³ − 4ab²

Step 1: Factor out the common factor:
a³ − 4ab² = a(a² − 4b²)
Step 2: Recognize the difference of squares:
a² − 4b² = a² − (2b)²
Step 3: Apply the identity A² − B² = (A − B)(A + B):
a² − (2b)² = (a − 2b)(a + 2b)
Step 4: Write the complete factorisation:
a³ − 4ab² = a(a − 2b)(a + 2b)
Answer: a³ − 4ab² = a(a − 2b)(a + 2b)

Q29: ab³c − abc³

Step 1: Factor out the common factor:
ab³c − abc³ = abc(b² − c²)
Step 2: Recognize the difference of squares:
b² − c² = (b − c)(b + c)
Step 3: Write the complete factorisation:
ab³c − abc³ = abc(b − c)(b + c)
Answer: ab³c − abc³ = abc(b − c)(b + c)

Q30: 9(x+y)³ − 16(x+y)

Step 1: Factor out the common factor (x+y):
9(x+y)³ − 16(x+y) = (x+y)(9(x+y)² − 16)
Step 2: Recognize the difference of squares:
9(x+y)² − 16 = (3(x+y))² − 4²
Step 3: Apply the identity A² − B² = (A − B)(A + B):
(3(x+y))² − 4² = (3(x+y) − 4)(3(x+y) + 4) = (3x + 3y − 4)(3x + 3y + 4)
Step 4: Write the complete factorisation:
9(x+y)³ − 16(x+y) = (x + y)(3x + 3y − 4)(3x + 3y + 4)
Answer: (x + y)(3x + 3y − 4)(3x + 3y + 4)

Q31: 1 − 0.49c⁶

Step 1: Express the decimal as a square:
0.49 = 0.7² → 1 − 0.49c⁶ = 1 − (0.7c³)²
Step 2: Apply the difference of squares identity A² − B² = (A − B)(A + B):
1 − (0.7c³)² = (1 − 0.7c³)(1 + 0.7c³)
Answer: (1 − 0.7c³)(1 + 0.7c³)

Q32: x² − y² − 8yz − 16z²

Step 1: Rearrange terms for grouping:
x² − (y² + 8yz + 16z²)
Step 2: Recognize the perfect square:
y² + 8yz + 16z² = (y + 4z)² → x² − (y + 4z)²
Step 3: Apply the difference of squares identity A² − B² = (A − B)(A + B):
x² − (y + 4z)² = (x − (y + 4z))(x + (y + 4z))
Step 4: Simplify the factors:
(x − (y + 4z)) = x − y − 4z
(x + (y + 4z)) = x + y + 4z
Answer: (x − y − 4z)(x + y + 4z)

Q33: \(x^3y^3 – \frac{25xy}{z^2}\)

Step 1: Find the common factor of both terms: \[ = xy\left(x^2y^2 – \frac{25}{z^2}\right) \]Step 2: Recognize that \(x^2y^2 – \frac{25}{z^2}\) is a difference of squares: \[ x^2y^2 = (xy)^2,\quad \frac{25}{z^2} = \left(\frac{5}{z}\right)^2 \]Step 3: Apply the identity \(a^2 – b^2 = (a – b)(a + b)\): \[ = xy\left(xy – \frac{5}{z}\right)\left(xy + \frac{5}{z}\right) \]Answer: xy\left(xy – \frac{5}{z}\right)\left(xy + \frac{5}{z}\right)

Q34: \(0.0324x^4 – 0.0064b^4\)

Step 1: Find the common factor of both terms: \[ 0.0324x^4 – 0.0064b^4 = 4(0.0081x^4 – 0.0016b^4) \] Step 2: Express the decimals as squares: \[ 0.0081 = (0.09)^2,\quad 0.0016 = (0.04)^2 \] Thus, \[ = 4 (0.09x^2)^2 – (0.04b^2)^2 \]Step 2: Apply the difference of squares formula \(a^2 – b^2 = (a – b)(a + b)\): \[ = 4\left(0.09x^2 – 0.04b^2\right)\left(0.09x^2 + 0.04b^2\right) \]Step 3: Factor each term further if possible.
First factor: \[ 0.09x^2 – 0.04b^2 = (0.3x – 0.2b)(0.3x + 0.2b) \]Step 4: Combine the factored terms: \[ = 4(0.3x – 0.2b)(0.3x + 0.2b)\times (0.09x^2 + 0.04b^2) \\ = 4(0.09x^2 + 0.04b^2)(0.3x + 0.2b)(0.3x – 0.2b) \]Answer: \(4(0.09x^2 + 0.04b^2)(0.3x + 0.2b)(0.3x – 0.2b)\)

Q35: Using the identity \(\left(a^2-b^2\right)=(a+b)(a-b)\), evaluate each of the following:

i. 82² − 18²

Step 1: Apply the identity: (82 + 18)(82 − 18)
Answer: 6400

ii. (15.8)² − (9.2)²

Step 1: (15.8 + 9.2)(15.8 − 9.2)
Answer: 165

iii. (0.8)² − (0.2)²

Step 1: (0.8 + 0.2)(0.8 − 0.2)
Answer: 0.6

iv. \(\left(7\frac{3}{4}\right)^2-\left(2\frac{1}{4}\right)^2\)

Step 1: Convert to improper fractions: 31/4, 9/4
Step 2: Apply identity: \((\frac{31}{4} + \frac{9}{4}) (\frac{31}{4} − \frac{9}{4}) = 10 × 5.5\)
Answer: 55

v. \(\left(6\frac{4}{11}\right)^2-\left(4\frac{7}{11}\right)^2\)

Step 1: Convert to improper fractions: 70/11, 51/11
Step 2: Apply identity: \((\frac{70}{11} + \frac{51}{11}) (\frac{70}{11} − \frac{51}{11}) = 11 × \frac{19}{11}\)
Answer: 19

vi. \(\frac{7.3\times7.3-2.7\times2.7}{7.3-2.7}\)

Step 1: Numerator: 7.3² − 2.7² = (7.3 + 2.7)(7.3 − 2.7)
Step 2: Divide by (7.3 − 2.7)
→ \(\frac{(7.3 + 2.7)(7.3 − 2.7)}{(7.3 − 2.7)} = (7.3 + 2.7) = 10\)
Answer: 10