**Factorize**

**(i) x**^{2}**+ 9x + 18 **

Solution:

= x^{2} + 6x + 3x + 18

= x (x + 6) + 3 (x + 6)

= (x + 6) (x + 3)

**(ii) y**^{2}**+ 5y – 24 **

Solution:

= y^{2} + 8y – 3y – 24

= y (y + 8) – 3 (y + 8)

= (y + 8) (y – 3)

**(iii) 7y**^{2}**+ 49y +84 **

Solution:

= 7 (y^{2} + 7y + 12)

= 7 (y^{2} + 4y + 3y + 12)

=7 [y (y + 4) + 3 (y + 4)]

=7 (y + 4) (y + 3)

**(iv) 40 + 3x – x**^{2}

Solution:

= – [x^{2} – 3x – 40]

= 3 – [x^{2} – 8x + 5x – 40]

= – [x (x – 8) + 5 (x – 8)]

= [(x – 8) (x + 5)]

= (8 – x) (x + 5)

**(v) m**^{2}**+ 17mn – 84n**^{2 }

Solution:

= m^{2} + 21mn – 4mn – 84n^{2}

= m (m + 21n) – 4n (m + 21n)

= (m + 21m) (m – 4n)

**(vi) 117p**^{2 }**+ 2pq – 24q**^{2 }

Solution:

= 117p^{2} + 54pq – 52pq – 24q^{2}

= 9p (13p + 6q) – 3q (13p + 6q)

= (9p – 3q) (13p + 6q)

**(vii) 15x**^{2}**– r – 28 **

Solution:

= 15x^{2} – 21x + 20x – 28

= 3x (5x – 7) + 4 (5x – 7)

= (3x + 4) (5x – 7)

**(viii) 2x**^{2}**– x – 21 **

Solution:

= 2x^{2} – 7x + 6x – 21

= x (2x – 7) + 3 (2x – 7)

= (2x – 7) (x + 3)

**(ix) 8k**^{2}**– 22k – 21 **

Solution:

= 8k^{2} – 28k + 6k – 21

= 4k (2k – 7) + 3(2k – 7)

= (2k – 7) (4k + 3)

**(x) **^{𝟏}**/ _{𝟑} **

**x**

^{2}**– 2x – 9**

Solution:

= (x^{2}+ 6x−27)/_{3}

= ^{1}/_{3} [x^{2} – 6x – 27]

= ^{1}/_{3} [x^{2} – 9x + 3x – 27]

= ^{1}/_{3} [x (x – 9) + 3 (x – 9)]

=^{1}/_{3} (x – 9) (x + 3)

**Factorize**

**(i) √****𝟓 ****x**^{2}**+ 2x – 3√****𝟓 **

Solution:

= √5 x2 + 5x – 3x – 3√5

= √5 x (x + √5 ) – 3 ((x +√ 5 )

= (x + √5 ) (√5 x – 3)

**(ii) √****𝟑 ****a**^{2}**+ 2a – 5√****𝟑 **

Solution:

= √3a^{2} + 5a – 3a – 5√3 – 5√3 – √3

= a (√3 a + 5) – √3 (√3 a + 5)

= (√3 a + 5) (a – √3 )

**(iii) 7√****2 ****y**^{2}**– 10y – 4√****2 **

Solution:

= 7√2 y^{2} – 14y + 4y – 4√2

= 7√2 y (y – √2) + 4 (y – 2)

= (y – √2) (7√2 y + 4)

**(iv) 6√****𝟑 ****z**^{2}**– 47z – 5√****𝟑 **

Solution:

= 6√3 z^{2 }– 45z – 2z – 5√3

= 3√3 z (2z –5√3) – (2z – 5√3)

= (2z – 5√3) (3√3z –1)

**(v) 4√****𝟑****x**^{2}**+ 5x – 2√****𝟑 **

Solution:

= 4√3 x^{2} + 8x – 3x – 2√3

= 4x (√3x + 2) – √3(√3x + 2)

= (√3x + 2) (4x –√3)

**Factorize**

**(i) 2 (x + y)**^{2}**– 9 (x + y) – 5 **

Solution:

Let x + y = p

= 2p^{2} – 9p – 5

= 2p^{2} – 10p + p – 5

= 2p(p – 5) + 1 (p – 5)

= (p – 5) (2p + 1)

= (x + y – 5) [2(x + y) + 1]

= (x + y – 5) (2x + 2y + 1)

**(ii) 2(a – 2b)**^{2}**– 25(a – 2b) + 12 **

Solution:

Put a – 2b = x

= 2x^{2} – 25x + 12

= 2x^{2} – 24x – x + 12

= 2x (x – 12) – 1 (x – 12)

= (x – 12) (2x – 1)

= (a – 2b – 12) [2(a – 2b) – 1]

= (a – 2b – 12) [2a – 4b – 1]

**(iii) 12(z + 1)**^{2}**– 25(z + 1) (x + 2) + 12 (x + 2)**^{2}

Solution:

Let z + 1 = a x + 2 = b

= 12a^{2} – 25ab + 12b^{2}

= 12a^{2} – 16ab – 9ab + 12b^{2}

= 4a (3a – 4b) – 3b (3a – 4b)

= (3a – 4b) (4a – 3b)

= [3 (z + 1) – 4 (x + 2)] [4 (z + 1) – 3 (x + 2)]

= (3z + 3 – 4x – 8) (4z + 4 3x – 2)

= (3z – 4x – 5) (4z – 3x – 2)

**(iv) 9(2x – y) – 4(2x – y) – 13 **

Solution:

9(2x – y) – 4(2x – y) – 13

Put 2x – y = a

= 9a + 9a – 13a – 13

= 9a (a + 1) – 13 (a +1)

= (a + 1) (9a – 13)

= (2x – y + 1) [9 (2x – y) – 13]

= (2x – y + 1) (18x – 9y – 13)

**Factorize**

**(i) x**^{4}**– 3x**^{2}**+ 2 **

Solution:

Put x^{2} = a

a^{2} – 3a + 2

= a^{2} – 2a – a + 2

= a (a – 2) – 1 (a – 2)

= (a – 2) (a – 1)

= (x^{2} – 2) (x^{2} – 1)

= (x – 2 ) (x + 2 ) (x – 1) (x + 1) [∵a^{2} – b^{2} = (a + b) (a – b)]

**(ii) 4x**^{4}**+ 7x**^{2}**– 2 **

Solution:

Put x^{2} = a

= 4a^{2} + 7a – 2

= 4a^{2} + 8a – a – 2

= 4a (a + 2) – 1 (a + 2)

= (a + 2) (4a – 1) [∵a^{2 }– b^{2} = (a + b) (a – b)]

= (x^{2} + 2) (4x^{2} – 1)

= (x^{2} + 2) (2x + 1) (2x – 1)

**(iii) 3x**^{3}**– x**^{2}**– 10x **

Solution:

= x (3x^{2} – x – 10)

= x (3x^{2} – 6x + 5x – 10)

= x [3x (x – 2) + 5 (x – 2)]

= x (x – 2) (3x + 5)

** **

**(iv) 8x**^{3}**+ 2x**^{2}**y – 15xy**^{2}

Solution:

= x (8x^{2} – 2xy – 15y^{2})

= x (8x^{2} – 12xy + 10xy – 15y^{2})

= x [4x (2x – 3y) + 5y (2x – 3y)]

= x (2x – 3y) (4x + 5y)

** **

**(v) x**^{6}**– 7x**^{3}**– 8 **

Solution:

Put x^{3} = a

a^{3} – 7a – 8 [∵a^{3} + b^{3} = (a + b) (a^{2 }– ab + b^{2})]

= a^{3} – 8a + a – 8 [∵a^{3} – b^{3} = (a – b) (a^{2} – ab + b^{2})]

= a (a – 8) + 1 (a – 8)

= (a – 8) (a + 1)

= (x^{3} – 3) (x^{3} + 1)

= (x^{3 }– 3) (x^{3} + 1)

= (x + 1) (x^{2} – x + 1) (x – 2) (x^{2} + 2x + 4)

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