Factorisation

Factorization is the process of breaking down an expression into simpler expressions or factors, which when multiplied together produce the original expression.

Key Concepts:

Factors: Numbers or algebraic expressions that divide a given number or expression exactly.
Methods of Factorization:
- Common Factors: Finding the common factor in all terms and taking it out.
- Factorization by Grouping: Grouping terms to find common factors.
- Factorization using Identities: Applying algebraic identities like (a + b)², (a – b)², and a² – b².
- Division of Algebraic Expressions: Dividing polynomials by monomials or binomials.

Importance:

Factorization is crucial for simplifying expressions, solving equations, and understanding algebraic concepts. It’s a fundamental tool in higher-level mathematics.

Example: Factorize 6x² + 11x + 3

We can factorize this expression as (2x + 3)(3x + 1)

By understanding factorization, you can effectively manipulate algebraic expressions and solve various mathematical problems.

Exercise 12.1

1. Find the common factors of the given terms.

(i) 12x, 36

(ii) 2y, 22xy

(iii) 14pq, 28p2q2

(iv) 2x, 3x2, 4

(v) 6abc, 24ab2, 12a2b

(vi) 16x3, -4x2, 32x

(vii) 10pq, 20qr, 30rp

(viii) 3x2y3, 10x3y2, 6x2y2z

Ans :

(i) 12x, 36

Factors of 12x: 1, 2, 3, 4, 6, 12, x
Factors of 36:
1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors:
1, 2, 3, 4, 6, 12

(ii) 2y, 22xy

Factors of 2y: 1, 2, y
Factors of 22xy: 1, 2, 11, 22, x, y
Common factors: 1, 2, y

(iii) 14pq, 28p²q²

Factors of 14pq: 1, 2, 7, 14, p, q
Factors of 28p²q²: 1, 2, 4, 7, 14, 28, p, p, q, q
Common factors: 1, 2, 7, p, q

(iv) 2x, 3x², 4

Factors of 2x: 1, 2, x
Factors of 3x²: 1, 3, x, x
Factors of 4: 1, 2, 4
Common factor: 1

(v) 6abc, 24ab², 12a²b

Factors of 6abc: 1, 2, 3, 6, a, b, c
Factors of 24ab²: 1, 2, 3, 4, 6, 8, 12, 24, a, b, b
Factors of 12a²b: 1, 2, 3, 4, 6, 12, a, a, b
Common factors: 1, 2, 3, 6, a, b

(vi) 16x³, -4x², 32x

Factors of 16x³: 1, 2, 4, 8, 16, x, x, x
Factors of -4x²: -1, 1, 2, 4, x, x
Factors of 32x: 1, 2, 4, 8, 16, 32, x
Common factors: 1, 2, 4, x

(vii) 10pq, 20qr, 30rp

Factors of 10pq: 1, 2, 5, 10, p, q
Factors of 20qr: 1, 2, 4, 5, 10, 20, q, r
Factors of 30rp: 1, 2, 3, 5, 6, 10, 15, 30, r, p
Common factors: 1, 2, 5

(viii) 3x²y³, 10x³y², 6x²y²z

Factors of 3x²y³: 1, 3, x, x, y, y, y
Factors of 10x³y²: 1, 2, 5, 10, x, x, x, y, y
Factors of 6x²y²z: 1, 2, 3, 6, x, x, y, y, z
Common factors: 1, x, x, y, y

2. Factorise the following expressions.

(i) 7x – 42

(ii) 6p – 12q

(iii) 7a2 + 14a

(iv) -16z + 20z3

(v) 20l2m + 30alm

(vi) 5x2y – 15xy2

(vii) 10a2 – 15b2 + 20c2

(viii) -4a2 + 4ab – 4ca

(ix) x2yz + xy2z + xyz2

(x) ax2y + bxy2 + cxyz

Ans :

Let’s factorize the given expressions:

Common Factor Method:

(i) 7x – 42

Common factor is 7
Factorized form: 7(x – 6)

(ii) 6p – 12q

Common factor is 6
Factorized form: 6(p – 2q)

(iii) 7a² + 14a

Common factor is 7a
Factorized form: 7a(a + 2)

(iv) -16z + 20z³

Common factor is -4z
Factorized form: -4z(4 – 5z²)

(v) 20l²m + 30alm

Common factor is 10lm
Factorized form: 10lm(2l + 3a)

(vi) 5x²y – 15xy²

Common factor is 5xy
Factorized form: 5xy(x – 3y)

Grouping Method:

(vii) 10a² – 15b² + 20c²

There is no common factor for all terms.
This expression cannot be factored further.

(viii) -4a² + 4ab – 4ca

Common factor is -4a
Factorized form: -4a(a – b + c)

Taking Common Factors:

(ix) x²yz + xy²z + xyz²

Common factor is xyz
Factorized form: xyz(x + y + z)

(x) ax²y + bxy² + cxyz

Common factor is xyz
Factorized form: xyz(a + by + cz)

3. Factorise:

(i) x2 + xy + 8x + 8y

(ii) 15xy – 6x + 5y – 2

(iii) ax + bx – ay – by

(iv) 15pq + 15 + 9q + 25p

(v) z – 7 + 7xy – xyz

Ans :

Applying Factorization Techniques

i) x² + xy + 8x + 8y

Group the terms: (x² + xy) + (8x + 8y)
Factor out the common terms in each group: x(x + y) + 8(x + y)
Factor out the common binomial: (x + y)(x + 8)

ii) 15xy – 6x + 5y – 2

Group the terms: (15xy – 6x) + (5y – 2)
Factor out the common terms in each group: 3x(5y – 2) + 1(5y – 2)
Factor out the common binomial: (5y – 2)(3x + 1)

iii) ax + bx – ay – by

Group the terms: (ax + bx) – (ay + by)
Factor out the common terms in each group: x(a + b) – y(a + b)
Factor out the common binomial: (a + b)(x – y)

iv) 15pq + 15 + 9q + 25p

Rearrange the terms: 15pq + 25p + 9q + 15
Group the terms: (15pq + 25p) + (9q + 15)
Factor out the common terms in each group: 5p(3q + 5) + 3(3q + 5)
Factor out the common binomial: (3q + 5)(5p + 3)

v) z – 7 + 7xy – xyz

Rearrange the terms: z – xyz + 7xy – 7
Group the terms: z(1 – xy) + 7(xy – 1)
Factor out the common terms in each group: z(1 – xy) – 7(1 – xy)
Factor out the common binomial: (1 – xy)(z – 7)

Exercise 12.2

1. Factorise the following expressions.

(i) a2 + 8a +16

(ii) p2 – 10p + 25

(iii) 25m2 + 30m + 9

(iv) 49y2 + 84yz + 36z2

(v) 4x2 – 8x + 4

(vi) 121b2 – 88bc + 16c2

(vii) (l + m)2 – 4lm. (Hint: Expand (l + m)2 first)

(viii) a4 + 2a2b2 + b4

Ans :

These expressions are perfect squares trinomials, which can be factored using the formula (a + b)² = a² + 2ab + b² or (a – b)² = a² – 2ab + b².

(i) a² + 8a + 16

a² + 8a + 16 = (a + 4)²

(ii) p² – 10p + 25

Here, a = p and b = 5
So, p² – 10p + 25 = (p – 5)²

(iii) 25m² + 30m + 9

Here, a = 5m and b = 3
So, 25m² + 30m + 9 = (5m + 3)²

(iv) 49y² + 84yz + 36z²

Here, a = 7y and b = 6z
So, 49y² + 84yz + 36z² = (7y + 6z)²

(v) 4x² – 8x + 4

First, factor out the common factor 4: 4(x² – 2x + 1)
Now, factorize the quadratic: 4(x – 1)²

(vi) 121b² – 88bc + 16c²

121b² – 88bc + 16c² = (11b – 4c)²

(vii) (l + m)² – 4lm

Expand (l + m)²: l² + 2lm + m² – 4lm
Combine like terms: l² – 2lm + m²
Factorize as a difference of two squares: (l – m)²

(viii) a⁴ + 2a²b² + b⁴

Here, a = a² and b = b²
So, a⁴ + 2a²b² + b⁴ = (a² + b²)²

2. Factorise.

(i) 4p2 – 9q2

(ii) 63a2 – 112b2

(iii) 49x2 – 36

(iv) 16x5 – 144x3

(v) (l + m)2 – (l – m)2

(vi) 9x2y2 – 16

(vii) (x2 – 2xy + y2) – z2

(viii) 25a2 – 4b2 + 28bc – 49c2

Ans :

Applying Factorization Techniques

These expressions involve the difference of two squares, which can be factored using the formula: a² – b² = (a + b)(a – b)

i) 4p² – 9q²

Here, a = 2p and b = 3q
So, 4p² – 9q² = (2p + 3q)(2p – 3q)

ii) 63a² – 112b²

First, factor out the common factor 7: 7(9a² – 16b²)
Now, factorize the difference of two squares: 7(3a + 4b)(3a – 4b)

iii) 49x² – 36

Here, a = 7x and b = 6
So, 49x² – 36 = (7x + 6)(7x – 6)

iv) 16x⁵ – 144x³

First, factor out the common factor 16x³: 16x³(x² – 9)
Now, factorize the difference of two squares: 16x³(x + 3)(x – 3)

v) (l + m)² – (l – m)²

This is a difference of two squares where a = (l + m) and b = (l – m)
So, (l + m)² – (l – m)² = [(l + m) + (l – m)][(l + m) – (l – m)]
Simplify: (2l)(2m) = 4lm

vi) 9x²y² – 16

Here, a = 3xy and b = 4
So, 9x²y² – 16 = (3xy + 4)(3xy – 4)

vii) (x² – 2xy + y²) – z²

First, factorize the perfect square trinomial: (x – y)² – z²
Now, factorize the difference of two squares: (x – y + z)(x – y – z)

viii) 25a² – 4b² + 28bc – 49c²

Group the terms: (25a² – 4b²) + (28bc – 49c²)
Factorize the difference of two squares in the first group: (5a + 2b)(5a – 2b) + 7c(4b – 7c)
Notice that (5a – 2b) is common: (5a – 2b)(5a + 2b + 7c)

3. Factorise the expressions.

(i) ax2 + bx

(ii) 7p2 + 21q2

(iii) 2x3 + 2xy2 + 2xz2

(iv) am2 + bm2 + bn2 + an2

(v) (lm + l) + m + 1

(vi) y(y + z) + 9(y + z)

(vii) 5y2 – 20y – 8z + 2yz

(viii) 10ab + 4a + 5b + 2

(ix) 6xy – 4y + 6 – 9x

Ans :

Solutions:

i) ax² + bx

Common factor is x
Factorized form: x(a + b)

ii) 7p² + 21q²

Common factor is 7
Factorized form: 7(p² + 3q²)

iii) 2x³ + 2xy² + 2xz²

Common factor is 2x
Factorized form: 2x(x² + y² + z²)

iv) am² + bm² + bn² + an²

Group the terms: (am² + bm²) + (bn² + an²)
Factor out common terms: m²(a + b) + n²(a + b)
Factor out common binomial: (a + b)(m² + n²)

v) (lm + l) + m + 1

Rearrange: lm + l + m + 1
Group the terms: (lm + l) + (m + 1)
Factor out common terms: l(m + 1) + 1(m + 1)
Factor out common binomial: (m + 1)(l + 1)

vi) y(y + z) + 9(y + z)

Common factor is (y + z)
Factorized form: (y + z)(y + 9)

vii) 5y² – 20y – 8z + 2yz

Rearrange the terms: 5y² – 20y + 2yz – 8z
Group the terms: (5y² – 20y) + (2yz – 8z)
Factor out common terms: 5y(y – 4) + 2z(y – 4)
Factor out common binomial: (y – 4)(5y + 2z)

viii) 10ab + 4a + 5b + 2

Rearrange the terms: 10ab + 5b + 4a + 2
Group the terms: (10ab + 5b) + (4a + 2)
Factor out common terms: 5b(2a + 1) + 2(2a + 1)
Factor out common binomial: (2a + 1)(5b + 2)

ix) 6xy – 4y + 6 – 9x

Rearrange the terms: 6xy – 9x – 4y + 6
Group the terms: (6xy – 9x) – (4y – 6)
Factor out common terms: 3x(2y – 3) – 2(2y – 3)
Factor out common binomial: (2y – 3)(3x – 2)

4. Factorise.

(i) a4 – b4

(ii) p4 – 81

(iii) x4 – (y + z)4

(iv) x4 – (x – z)4

(v) a4 – 2a2b2 + b4

Ans :

These expressions involve the difference of two squares, which can be factored using the formula: a² – b² = (a + b)(a – b)

i) a⁴ – b⁴

This can be seen as a difference of two squares where a² = a⁴ and b² = b⁴
So, a⁴ – b⁴ = (a² + b²)(a² – b²)
Now, factorize the second term as a difference of two squares:
(a² + b²)(a + b)(a – b)

ii) p⁴ – 81

This can be seen as a difference of two squares where a² = p⁴ and b² = 81
So, p⁴ – 81 = (p² + 9)(p² – 9)
Now, factorize the second term as a difference of two squares:
(p² + 9)(p + 3)(p – 3)

iii) x⁴ – (y + z)⁴

This can be seen as a difference of two squares where a² = x⁴ and b² = (y + z)⁴
So, x⁴ – (y + z)⁴ = [x² + (y + z)²][x² – (y + z)²]
The first term cannot be factored further.
The second term is a difference of two squares:
[x² + (y + z)²][x + (y + z)][x – (y + z)]

iv) x⁴ – (x – z)⁴

This can be seen as a difference of two squares where a² = x⁴ and b² = (x – z)⁴
So, x⁴ – (x – z)⁴ = [x² + (x – z)²][x² – (x – z)²]
The first term cannot be factored further.
The second term is a difference of two squares:
[x² + (x – z)²][x + (x – z)][x – (x – z)]

v) a⁴ – 2a²b² + b⁴

This is a perfect square trinomial: (a² – b²)²

5. Factorise the following expressions.

(i) p2 + 6p + 8

(ii) q2 – 10q + 21

(iii) p2 + 6p – 16

Ans :

i) p² + 6p + 8

We need two numbers that multiply to 8 and add up to 6.
These numbers are 4 and 2.
So, p² + 6p + 8 = (p + 4)(p + 2)

ii) q² – 10q + 21

We need two numbers that multiply to 21 and add up to -10.
These numbers are -7 and -3.
So, q² – 10q + 21 = (q – 7)(q – 3)

iii) p² + 6p – 16

These numbers are 8 and -2.
So, p² + 6p – 16 = (p + 8)(p – 2)

Exercise 12.3

1. Carry out the following divisions.

(i) 28x4 ÷ 56x

(ii) -36y3 ÷ 9y2

(iii) 66pq2r3 ÷ 11qr2

(iv) 34x3y3z3 ÷ 51xy2z3

(v) 12a8b8 ÷ (-6a6b4)

Ans :

Solutions:

i) 28x⁴ ÷ 56x

Divide coefficients: 28 ÷ 56 = 1/2
Divide variables: x⁴ ÷ x = x^(4-1) = x³
Result: (1/2)x³

ii) -36y³ ÷ 9y²

Divide coefficients: -36 ÷ 9 = -4
Divide variables: y³ ÷ y² = y^(3-2) = y
Result: -4y

iii) 66pq²r³ ÷ 11qr²

Divide coefficients: 66 ÷ 11 = 6
Divide variables: p / p = 1, q² / q = q^(2-1) = q, r³ / r² = r^(3-2) = r
Result: 6pq

iv) 34x³y³z³ ÷ 51xy²z³

Divide coefficients: 34 ÷ 51 = 2/3
Divide variables: x³ / x = x^(3-1) = x², y³ / y² = y^(3-2) = y, z³ / z³ = 1
Result: (2/3)x²y

v) 12a⁸b⁸ ÷ (-6a⁶b⁴)

Divide coefficients: 12 ÷ (-6) = -2
Divide variables: a⁸ / a⁶ = a^(8-6) = a², b⁸ / b⁴ = b^(8-4) = b⁴
Result: -2a²b⁴

2. Divide the following polynomial by the given monomial.

(i) (5x2 – 6x) ÷ 3x

(ii) (3y8 – 4y6 + 5y4) ÷ y4

(iii) 8(x3y2z2 + x2y3z2 + x2y2z3) ÷ 4x2y2z2

(iv) (x3 + 2x2 + 3x) ÷ 2x

(v) (p3q6 – p6q3) ÷ p3q3

Ans :

i) (5x² – 6x) ÷ 3x

Divide each term by 3x:
- (5x²/3x) – (6x/3x)
- Simplify: (5/3)x – 2

ii) (3y⁸ – 4y⁶ + 5y⁴) ÷ y⁴

Divide each term by y⁴:
- (3y⁸/y⁴) – (4y⁶/y⁴) + (5y⁴/y⁴)
- Simplify: 3y⁴ – 4y² + 5

iii) 8(x³y²z² + x²y³z² + x²y²z³) ÷ 4x²y²z²

First, distribute the 8: 8x³y²z² + 8x²y³z² + 8x²y²z³
Then divide each term by 4x²y²z²:
- (8x³y²z²/4x²y²z²) + (8x²y³z²/4x²y²z²) + (8x²y²z³/4x²y²z²)
- Simplify: 2x + 2y + 2z

iv) (x³ + 2x² + 3x) ÷ 2x

Divide each term by 2x:
- (x³/2x) + (2x²/2x) + (3x/2x)
- Simplify: (1/2)x² + x + (3/2)

v) (p³q⁶ – p⁶q³) ÷ p³q³

Divide each term by p³q³:
- (p³q⁶/p³q³) – (p⁶q³/p³q³)
- Simplify: q³ – p³

3. Work out the following divisions.

(i) (10x – 25) ÷ 5

(ii) (10x – 25) ÷ (2x – 5)

(iii) 10y(6y + 21) ÷ 5(2y + 7)

(iv) 9x2y2(3z – 24) ÷ 27xy(z – 8)

(v) 96abc(3a – 12) (5b – 30) ÷ 144(a – 4)(b – 6)

Ans :

Factor out 5 from the numerator: 5(2x – 5) ÷ 5
Cancel out the common factor 5: 2x – 5

ii)

The numerator can be factored as 5(2x – 5)
The expression becomes: 5(2x – 5) / (2x – 5)
Cancel out the common factor (2x – 5): 5

iii)

Factor out 5 and 7 from the respective expressions: 5 * 2y(3y + 7) / 5(2y + 7)
Cancel out the common factors 5 and (2y + 7): 2y * 3 = 6y

iv)

Factor out 9, x, y, and 3 from the numerator and denominator: 9xy * x * y * 3(z – 8) / 9xy * 3(z – 8)
Cancel out the common factors: xy

Factor out common factors: 48 * 2abc * 3(a – 4) * 5(b – 6) / 48 * 3 * (a – 4) * (b – 6)
Cancel out the common factors: 2abc * 5 = 10abc

4. Divide as directed.

(i) 5(2x + 1) (3x + 5) ÷ (2x + 1)

(ii) 26xy (x + 5)(y – 4) ÷ 13x(y – 4)

(iii) 52pqr(p + q) (q + r) (r + p) ÷ 104pq(q + r)(r + p)

(iv) 20(y + 4)(y2 + 5y + 3) ÷ 5(y + 4)

(v) x(x + 1) (x + 2) (x + 3) ÷ x(x + 1)

Ans :

i) 5(2x + 1)(3x + 5) ÷ (2x + 1)

Cancel out the common factor (2x + 1):
- 5(3x + 5)
- 15x + 25

ii) 26xy(x + 5)(y – 4) ÷ 13x(y – 4)

Cancel out the common factors 13, x, and (y – 4):
- 2y(x + 5)

iii) 52pqr(p + q)(q + r)(r + p) ÷ 104pq(q + r)(r + p)

Cancel out the common factors 52, p, q, (q + r), and (r + p):
- r / 2

iv) 20(y + 4)(y² + 5y + 3) ÷ 5(y + 4)

Cancel out the common factors 5 and (y + 4):
- 4(y² + 5y + 3)

v) x(x + 1)(x + 2)(x + 3) ÷ x(x + 1)

Cancel out the common factors x and (x + 1):
- (x + 2)(x + 3)

5. Factorise the expressions and divide them as directed.

(i) (y2 + 7y + 10) ÷ (y + 5)

(ii) (m2 – 14m – 32) ÷ (m + 2)

(iii) (5p2 – 25p + 20) ÷ (p – 1)

(iv) 4yz(z2 + 6z – 16) ÷ 2y(z + 8)

(v) 5pq(p2 – q2) ÷ 2p(p + q)

(vi) 12xy(9x2 – 16y2) ÷ 4xy(3x + 4y)

(vii) 39y3(50y2 – 98) ÷ 26y2(5y + 7)

Ans :

i) (y² + 7y + 10) ÷ (y + 5)

Factorize the numerator: (y + 2)(y + 5)
Divide by (y + 5): (y + 2)(y + 5) / (y + 5)
Cancel common factors: y + 2

ii) (m² – 14m – 32) ÷ (m + 2)

Factorize the numerator: (m – 16)(m + 2)
Divide by (m + 2): (m – 16)(m + 2) / (m + 2)
Cancel common factors: m – 16

iii) (5p² – 25p + 20) ÷ (p – 1)

Factor out 5 from the numerator: 5(p² – 5p + 4)
Factorize the quadratic: 5(p – 1)(p – 4)
Divide by (p – 1): 5(p – 1)(p – 4) / (p – 1)
Cancel common factors: 5(p – 4)

iv) 4yz(z² + 6z – 16) ÷ 2y(z + 8)

Factorize the quadratic in the numerator: 4yz(z + 8)(z – 2) ÷ 2y(z + 8)
Cancel common factors: 2z(z – 2)

v) 5pq(p² – q²) ÷ 2p(p + q)

Factorize the difference of squares in the numerator: 5pq(p + q)(p – q) ÷ 2p(p + q)
Cancel common factors: (5/2)q(p – q)

vi) 12xy(9x² – 16y²) ÷ 4xy(3x + 4y)

Factorize the difference of squares in the numerator: 12xy(3x + 4y)(3x – 4y) ÷ 4xy(3x + 4y)
Cancel common factors: 3(3x – 4y)

vii) 39y³(50y² – 98) ÷ 26y²(5y + 7)

Factor out common factors and factorize the difference of squares: 3 * 13 * y * y * y * 2(5y + 7)(5y – 7) / 2 * 13 * y * y * (5y + 7)
Cancel common factors: 3y(5y – 7)