**Exercise 5.1**

**Question 1: Factorize x**

^{3}+ x – 3x^{2}– 3**Solution:**

^{3}+ x – 3x

^{2}– 3

^{3}– 3x

^{2}is x

^{2}and 1 is the common factor in x – 3.

^{2}(x – 3) + 1(x – 3)

^{2}+ 1)

^{3}+ x – 3x

^{2}– 3 is (x – 3) (x

^{2}+ 1).

**Question 2: Factorize a(a + b)**

^{3}– 3a^{2}b(a + b)**Solution:**

^{3}– 3a

^{2}b(a + b)

^{3}– 3a

^{2}b(a + b) is a(a + b).

^{2}– 3ab]

^{2}+ b

^{2}+ 2ab – 3ab]

^{2}+ b

^{2}– ab)

^{3}– 3a

^{2}b(a + b) is a(a + b) (a

^{2}+ b

^{2}– ab).

**Question 3: Factorize x(x**

^{3}– y^{3}) + 3xy(x – y)**Solution:**

^{3}– y

^{3}) + 3xy(x – y)

^{3}- b

^{3}= (a – b) (a

^{2}+ ab + b

^{2})

^{2}+ xy + y

^{2}) + 3xy(x – y)

^{2}+ xy + y

^{2}) + 3xy(x – y) is x(x - y).

^{2}+ xy + y

^{2}+ 3y)

^{3}– y

^{3}) + 3xy(x – y) is x(x – y) (x

^{2}+ xy + y

^{2}+ 3y).

**Question 4: Factorize a**

^{2}x^{2}+ (ax^{2}+ 1)x + a**Solution:**

^{2}x

^{2}+ (ax

^{2}+ 1)x + a

^{2}x

^{2}+ a + (ax

^{2}+ 1)x

^{2}+ 1) + x(ax

^{2}+ 1)

^{2}+ 1) (a + x)

^{2}x

^{2}+ (ax

^{2}+ 1)x + a is (ax

^{2}+ 1) (a + x).

**Question 5: Factorize x**

^{2}+ y – xy – x**Solution:**

^{2}+ y – xy – x

^{2}– x – xy + y

^{2}+ y – xy – x is (x – 1) (x – y).

**Question 6: Factorize x**

^{3}– 2x^{2}y + 3xy^{2}– 6y^{3}**Solution:**

^{3}– 2x

^{2}y + 3xy

^{2}– 6y

^{3}

^{2}(x – 2y) + 3y

^{2}(x – 2y)

^{2}+ 3y

^{2})

^{3}– 2x

^{2}y + 3xy

^{2}– 6y

^{3}is (x – 2y) (x

^{2}+ 3y

^{2}).

**Question 7: Factorize 6ab – b**

^{2}+ 12ac – 2bc**Solution:**

^{2}+ 12ac – 2bc

^{2}– 2bc (Taking common)

^{2}+ 12ac – 2bc is (b + 2c) (6a – b)

**Question 8: Factorize (x**

^{2}+ 1/x^{2}) – 4(x+1/x)+6**Solution:**

**Question 9: Factorize x(x – 2) (x – 4) + 4x – 8**

**Solution:**

^{2}–4x+4)

^{2}–2(x)(2)+ (2)

^{2}]

^{2 }= a

^{2}+2ab+b

^{2}

^{2 }

^{3 }

^{3}

**Question 10: Factorize (x + 2) (x**

^{2}+ 25) – 10x^{2}– 20x**Solution:**

^{2}+ 25) – 10x

^{2}– 20x

^{2}+ 25) – 10x (x + 2)

^{2}+ 25 – 10x)

^{2}– 10x + 25)

^{2}– 5x – 5x + 25)

^{2}

^{2}+ 25) – 10x

^{2}– 20x is (x+2) (x– 5)

^{2}

**Question 11: Factorize 2a**

^{2}+ 2√6ab + 3b^{2}**Solution:**

^{2}+ 2√6ab + 3b

^{2}

^{2}+ 2 × √2a × √3b + (√3b)

^{2}

^{2}= a

^{2}+ b

^{2}+ 2ab

^{2 }

^{2}+ 2√6ab + 3b

^{2}is (√2a+√3b)

^{2}

**Question 12: Factorize (a – b + c)**

^{2}+ (b – c + a)^{2}+ 2(a – b + c) (b – c + a)**Solution:**

^{2}+ (b – c + a)2 + 2(a – b + c) (b – c + a)

^{2}= x

^{2}+ y

^{2}+ 2xy

^{2}

^{2}

^{2}

^{2}

^{2}+ ( b – c + a)

^{2}+ 2(a – b + c) (b – c + a) is 4a

^{2}

**Question 13: Factorize a**

^{2}+ b^{2}+ 2(ab + bc + ca)**Solution:**

^{2}+ b

^{2}+ 2ab + 2bc + 2ca

^{2}+ y2 + 2xy

^{2}+ 2bc + 2ca

^{2}+ 2c(b + a)

^{2}+ 2c(a + b)…………………………(1)

^{2}+ b

^{2}+ 2ab + 2bc + 2ca is (a + b)(a + b + 2c)

**Question 14: Factorize 4(x-y)**

^{2}– 12(x – y)(x + y) + 9(x + y)^{2}**Solution:**

^{2}– 12ab + 9b

^{2}

^{2}– 6ab – 6ab + 9b

^{2}

^{2}…………………………….(1)

^{2 }

^{2}

^{2}

^{2}

^{2}– 12(x – y)(x + y) + 9(x + y)

^{2}is (x + 5y)

^{2}

**Question 15: Factorize a**

^{2}– b^{2}+ 2bc – c^{2}**Solution:**

^{2}– b

^{2}+ 2bc – c

^{2}

^{2}+ y

^{2}+ 2xy

^{2}– (b – c)

^{2}

^{2}- y

^{2}= (x + y)(x - y)

^{2}– b

^{2}+ 2bc – c

^{2}is (a + b – c) (a – b + c)

**Question 16: Factorize a**

^{2}+ 2ab + b^{2}– c^{2}**Solution:**

^{2}– c

^{2}

^{2}+ 2ab + b

^{2}) – c

^{2}

^{2}= x

^{2}+ y

^{2}+ 2xy

^{2}– (c)

^{2}

^{2}- y

^{2}= (x + y)(x - y)

^{2}+ 2ab + b

^{2}– c

^{2}is (a + b + c) (a + b – c).

**Exercise 5.2**

**Factorize each of the following expressions:**

**Question 1: p**

^{3}+ 27**Solution:**

^{3}+ 27

^{3}+ 3

^{3}

^{3}+ b

^{3}) = (a + b)(a

^{2}–ab + b

^{2})

^{3}+ 27 is (p + 3)(p² – 3p – 9).

**Question 2: y**

^{3}+ 125**Solution:**

^{3}+ 125

^{3}+ 53

^{3}+ b

^{3}) = (a + b)(a

^{2}–ab + b

^{2})

^{2}− 5y + 5

^{2})

^{2}− 5y + 25)

^{3}+ 125 is (y + 5) (y

^{2}− 5y + 25)

**Question 3: 1 – 27a**

^{3}**Solution:**

^{3}

^{3}+ b

^{3}) = (a + b)(a

^{2}–ab + b

^{2})

^{2})

^{2})

^{3}is (1 − 3a)(1 + 3a + 9a

^{2})

**Question 4: 8x**

^{3}y^{3}+ 27a^{3}**Solution:**

^{3}y

^{3}+ 27a

^{3}

^{3}+ (3a)

^{3}

^{3}+ b

^{3}) = (a + b)(a

^{2}–ab + b

^{2})

^{2}−2xy×3a+(3a)

^{2})

^{2}y

^{2}−6xya + 9a

^{2})

^{3}y

^{3}+ 27a

^{3}is (2xy+3a)(4x

^{2}y

^{2}−6xya + 9a

^{2})

**Question 5: 64a**

^{3}− b^{3}^{ }

**Solution:**

^{3}− b

^{3}

^{3}− b

^{3}

^{3}– b

^{3}= (a – b)(a

^{2}+ ab + b

^{2})

^{2}+ 4a×b + b

^{2})

^{2}+4ab+b

^{2})

^{3}− b

^{3}is (4a−b) (16a

^{2}+ 4ab + b

^{2})

**Question 6: (x**

^{3})/216 – 8y^{3}**Solution:**

**Question 7: 10x**

^{4}y – 10xy^{4}^{ }

**Solution:**

^{4}y – 10xy

^{4}

^{3}− y

^{3})

^{3}– b

^{3}= (a – b)(a

^{2}+ ab + b

^{2})

^{2}+ xy + y

^{2})

^{4}y – 10xy

^{4}is 10xy (x−y)(x

^{2}+ xy + y

^{2})

**Question 8: 54x**

^{6}y + 2x^{3}y^{4}**Solution:**

^{6}y + 2x

^{3}y

^{4}

^{3}y(27x

^{3}+y

^{3})

^{3}y((3x)

^{3}+ y

^{3})

^{3}+ b

^{3}= (a + b)(a

^{2}– ab + b

^{2})

^{3}y {(3x+y) ((3x)

^{2}−3xy+y

^{2})}

^{6}y + 2x

^{3}y

^{4}is 2x

^{3}y(3x+y)(9x

^{2}− 3xy + y

^{2}).

**Question 9: 32a**

^{3}+ 108b^{3}**Solution:**

^{3}+ 108b

^{3}

^{3}+ 27b

^{3})

^{3}+(3b)

^{3})

^{3}+ b

^{3}= (a + b)(a

^{2}– ab + b

^{2})

^{2}−2a×3b+(3b)

^{2})]

^{3}+ 108b

^{3}is 4(2a+3b)(4a

^{2}− 6ab + 9b

^{2})

**Question 10: (a−2b)**

^{3}− 512b^{3}^{ }

**Solution:**

^{3}− 512b

^{3}

^{3}−(8b)

^{3}

^{3}– b

^{3}= (a – b)(a

^{2}+ ab + b

^{2})

^{2}+ (a−2b)8b + (8b)

^{2}}

^{2}+ 4b

^{2}− 4ab + 8ab − 16b

^{2}+ 64b

^{2})

^{3}− 512b

^{3}is (a−10b)(a

^{2}+ 52b

^{2}+ 4ab)

**Question 11: (a+b)**

^{3}− 8(a−b)^{3}**Solution:**

^{3}− 8(a−b)

^{3}

^{3}− [2(a−b)]

^{3}

^{3}− [2a−2b]

^{3}

^{3}– y

^{3}= (x – y)(x

^{2}+ xy + y

^{2})]

^{2}+(a+b)(2a−2b)+(2a−2b)

^{2})

^{2}+b

^{2}+2ab+(a+b)(2a−2b)+(2a−2b)

^{2})

^{2}+b

^{2}+2ab+2a

^{2}−2ab+2ab−2b

^{2}+(2a−2b)

^{2})

^{2}+2ab−b

^{2}+(2a−2b)

^{2})

^{2}+2ab−b

^{2}+4a

^{2}+4b

^{2}−8ab)

^{2}+4a

^{2}−b

^{2}+4b

^{2}−8ab+2ab)

^{3}− 8(a−b)

^{3}is (3b−a)(7a

^{2}+3b

^{2}−6ab)

**Question 12: (x+2)**

^{3}+ (x−2)^{3}**Solution:**

^{3}+ (x−2)

^{3}

^{3}+ b

^{3}= (a + b)(a

^{2}– ab + b

^{2})]

^{2}−(x+2)(x−2)+(x−2)

^{2})

^{2}+4x+4−(x+2)(x−2)+x

^{2}−4x+4)

^{2}− b

^{2 }

^{2}+ 8 − (x

^{2}− 2

^{2}))

^{2}+8 − x

^{2}+ 4)

^{2}+ 12)

^{3}+ (x−2)

^{3}is 2x(x

^{2}+ 12)

**Exercise 5.3**

**Question 1: Factorize 64a**

^{3}+ 125b^{3}+ 240a^{2}b + 300ab^{2}**Solution:**

^{3}+ 125b

^{3}+ 240a

^{2}b + 300ab

^{2}

^{3}= a

^{3}+ b

^{3}+ 3a

^{2}b + 3ab

^{2}

^{3}+ (5b)

^{3}+ 3(4a)

^{2}(5b) + 3(4a)(5b)

^{2 }

^{3}

^{3}+ 125b

^{3}+ 240a

^{2}b + 300ab

^{2}is (4a+5b)

^{3}

**Question 2: Factorize 125x**

^{3}– 27y^{3}– 225x^{2}y + 135xy^{2}**Solution:**

^{3}– 27y

^{3}– 225x

^{2}y + 135xy

^{2}

^{3}= a

^{3}− b

^{3}− 3a

^{2}b + 3ab

^{2 }

^{3}−(3y)

^{3}−3(5x)

^{2}(3y) + 3(5x)(3y)

^{2}

^{3}

^{3}– 27y

^{3}– 225x

^{2}y + 135xy

^{2}is (5x − 3y)

^{3}

**Question 3: Factorize 8/27 x**

^{3}+ 1 +4/3 x^{2}+ 2x**Solution:**

**Question 4: Factorize 8x**

^{3}+ 27y^{3}+ 36x^{2}y + 54xy^{2}**Solution:**

^{3}+ 27y

^{3}+ 36x

^{2}y + 54xy

^{2}

^{3}= a

^{3}+ b

^{3}+ 3a

^{2}b + 3ab

^{2}

^{3}+ (3y)

^{3}+ 3×(2x)

^{2}×3y + 3×(2x)(3y)

^{2}

^{3}

^{3}+ 27y

^{3}+ 36x

^{2}y + 54xy

^{2}is (2x+3y)

^{3}

**Question 5: Factorize a**

^{3}− 3a^{2}b + 3ab^{2}− b^{3}+ 8**Solution:**

^{3}− 3a

^{2}b + 3ab

^{2}− b

^{3}+ 8

^{3}= a

^{3}− b

^{3}− 3a

^{2}b + 3ab

^{2}

^{3}+ 2

^{3}

^{3}+ b

^{3}=(a + b)(a

^{2}– ab + b

^{2})]

^{2}−(a−b) × 2 + 2

^{2})

^{2}+b

^{2}−2ab−2(a−b)+4)

^{2}+b

^{2}−2ab−2a+2b+4)

^{3}− 3a

^{2}b + 3ab

^{2}− b

^{3}+ 8 is (a−b+2)(a

^{2}+b

^{2}−2ab−2a+2b+4)

**Exercise 5.4**

**Factorize each of the following expressions:**

**Question 1: a**

^{3}+ 8b^{3}+ 64c^{3}− 24abc**Solution:**

^{3}+ 8b

^{3}+ 64c

^{3}− 24abc

^{3}+ (2b)

^{3}+ (4c)

^{3}− 3× a × 2b × 4c

^{3}+b

^{3}+c

^{3}−3abc = (a + b + c) (a

^{2}+b

^{2}+c

^{2}−ab−bc−ca)

^{2}+(2b)

^{2}+ (4c)

^{2}−a× 2b − 2b × 4c − 4c × a)

^{2}+4b

^{2}+16c

^{2}−2ab−8bc−4ac)

^{3}+ 8b

^{3}+ 64c

^{3}− 24abc is (a+2b+4c) (a

^{2}+4b

^{2}+16c

^{2}−2ab−8bc−4ac)

**Question 2: x**

^{3}− 8y^{3}+ 27z^{3}+ 18xyz**Solution:**

^{3}− 8y

^{3}+ 27z

^{3}+ 18xyz

^{3}− (2y)

^{3}+ (3z)

^{3}– 3×x× (−2y)(3z)

^{2}+ (−2y)

^{2}+ (3z)

^{2}−x(−2y) − (−2y)(3z) −3z(x))

^{3}+ b

^{3}+ c

^{3}− 3abc = (a + b + c) (a

^{2}+ b

^{2}+ c

^{2}– ab – bc − ca)

^{2}+ 4y

^{2}+ 9z

^{2}+ 2xy + 6yz − 3zx)

^{3}− 8y

^{3}+ 27z

^{3}+ 18xyz is (x −2y + 3z) (x

^{2}+ 4y

^{2}+ 9z

^{2}+ 2xy + 6yz − 3zx)

**Question 3: 27x**

^{3}− y^{3}– z^{3}– 9xyz**Solution:**

^{3}− y

^{3}– z

^{3}– 9xyz

^{3}− y

^{3}– z

^{3}– 3(3xyz)

^{3}+ b

^{3}+ c

^{3}−3abc = (a + b + c)(a

^{2}+b

^{2}+c

^{2}−ab−bc−ca)]

^{2}+ (- y)

^{2}+ (– z)

^{2}+ 3xy – yz + 3xz)}

^{2}+ y

^{2}+ z

^{2}+ 3xy – yz + 3xz)}

^{3}− y

^{3}– z

^{3}– 9xyz is (3x – y – z) {9x

^{2}+ y

^{2}+ z

^{2}+ 3xy – yz + 3xz)}

**Question 4: 1/(27x**

^{3}) − y^{3}+ 125z^{3}+ 5xyz**Solution:**

**Question 5: 8x**

^{3}+ 27y^{3}− 216z^{3}+ 108xyz**Solution:**

^{3}+ 27y

^{3}− 216z

^{3}+ 108xyz

^{3}+ (3y)

^{3}+(−6y)

^{3}−3(2x)(3y)(−6z)

^{3}+ b

^{3}+ c

^{3}−3abc = (a + b + c)(a

^{2}+b

^{2}+c

^{2}−ab−bc−ca)]

^{2}+ (3y)

^{2}+ (−6z)

^{2}−2x × 3y − 3y × (−6z) − (−6z) × 2x }

^{2}+36z

^{2}−6xy + 18yz + 12zx}

^{3}+ 27y

^{3}− 216z

^{3}+ 108xyz is (2x +3y − 6z) {4x 2 +9y

^{2}+36z

^{2}−6xy + 18yz + 12zx}

**Question 6: 125 + 8x**

^{3}− 27y^{3}+ 90xy**Solution:**

^{3}− 27y

^{3}+ 90xy

^{3}+ (2x)

^{3}+(−3y)

^{3}−3 × 5 × 2x ×(−3y)

^{3}+ b

^{3}+ c

^{3}−3abc = (a + b + c)(a

^{2}+b

^{2}+c

^{2}−ab−bc−ca)]

^{2}+(2x)

^{2}+(−3y)

^{2}−5(2x)−2x(−3y) − (−3y)5)

^{2}+9y

^{2}− 10x + 6xy + 15y)

^{3}− 27y

^{3}+ 90xy is (5 + 2x − 3y)(25 + 4x

^{2}+9y

^{2}− 10x + 6xy + 15y)

**Question 7: (3x−2y)**

^{3}+ (2y−4z)^{3}+ (4z−3x)^{3}^{ }

**Solution:**

^{3}+ (2y−4z)

^{3}+ (4z−3x)

^{3}

^{3}+ b

^{3}+ c

^{3}−3abc = (a + b + c)(a

^{2}+b

^{2}+c

^{2}−ab−bc−ca)]

^{3}+ b

^{3}+ c

^{3}− 3abc = 0 (a

^{2}+b

^{2}+c

^{2}−ab−bc−ca)]

^{3}+ b

^{3}+ c

^{3}= 3abc

^{3}+ (2y−4z)

^{3}+ (4z−3x)

^{3}= 3(3x−2y)(2y−4z)(4z−3x)

^{3}+ (2y−4z)

^{3}+ (4z−3x)

^{3}is 3(3x−2y)(2y−4z)(4z−3x).

**Question 8: (2x−3y)**

^{3}+ (4z−2x)^{3}+ (3y−4z)^{3}**Solution:**

^{3}+ (4z−2x)

^{3}+ (3y−4z)

^{3}Let us assumed that

^{3}+ b

^{3}+ c

^{3}−3abc = (a + b + c)(a

^{2}+b

^{2}+c

^{2}−ab−bc−ca)]

^{3}+ b

^{3}+ c

^{3}− 3abc = 0 (a

^{2}+b

^{2}+c

^{2}−ab−bc−ca)]

^{3}+ b

^{3}+ c

^{3}= 3abc

^{3}+ (4z−2x)

^{3}+ (3y−4z)

^{3}= 3(2x−3y)(4z−2x)(3y−4z)

^{3}+(4z−2x)

^{3}+(3y−4z)

^{3}is (2x−3y)

^{3}+ (4z−2x)

^{3}+(3y−4z)

^{3}= 3(2x−3y)(4z−2x) (3y−4z)

**Exercise Very Short Answer Questions :->**

**Question 1: Factorize x**

^{4}+ x^{2}+ 25**Solution:**

^{4}+ x

^{2}+ 25

^{2})

^{2}+ 5

^{2}+ x

^{2}

^{2}+ b

^{2}= (a + b)

^{2}– 2ab

^{2}+ 5)

^{2}−2(x

^{2}) (5) + x

^{2}

^{2}+ 5)

^{2}−10x

^{2}+ x

^{2}

^{2}+ 5)

^{2}− 9x

^{2}

^{2}+ 5)

^{2}− (3x)

^{2}

^{2}– b

^{2}= (a + b) (a – b)

^{2}+ 5 + 3x) (x

^{2}+ 5 − 3x)

^{4}+ x

^{2}+ 25 is (x

^{2}+ 5 + 3x) (x

^{2}+ 5 − 3x).

**Question 2: Factorize x**

^{2}– 1 – 2a – a^{2}**Solution:**

^{2}– 1+2a+a

^{2}

^{2}– ((1)

^{2}+2a+a

^{2})

^{2}= a

^{2}+ b

^{2}+ 2ab

^{2}– (a + 1)

^{2}

^{2}– b

^{2}= (a + b) (a – b)

^{2}– 1 – 2a – a

^{2}is (x + a + 1) (x - a - 1)

**Question 3: If a + b + c = 0, then write the value of a**

^{3}+ b^{3}+ c^{3}.

**Solution:**

^{3}+ b

^{3}+ c

^{3}−3abc = (a + b + c)(a

^{2}+b

^{2}+c

^{2}−ab−bc−ca)]

^{3}+ b

^{3}+ c

^{3}– 3abc = (a + b +c) (a

^{2}+ b

^{2}+ c

^{2}– ab – bc − ca)

^{3}+ b

^{3}+ c

^{3}– 3abc = 0 (a

^{2}+ b

^{2}+ c

^{2}– ab – bc − ca)

^{3}+ b

^{3}+ c

^{3}– 3abc = 0

^{3}+ b

^{3}+ c

^{3}= 3abc

^{3}+ b

^{3}+ c

^{3}is 3abc.

**Question 4: If a**

^{2}+ b^{2}+ c^{2}= 20 and a + b + c = 0, find ab + bc + ca.

**Solution:**

^{2}+ b

^{2}+ c

^{2}= 20 and a + b + c = 0

^{2}+ b

^{2}+ c

^{2}and a + b + c in equation (1)

**Question 5: If a + b + c = 9 and ab + bc + ca = 40, find a**

^{2}+ b^{2}+ c^{2}**Solution:**

^{2}= a² + b² + c² + 2(40)