1. Factorize

(i) x² + 9x + 18

Solution:

= x² + 6x + 3x + 18

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

= (x + 6) (x + 3)

 

(ii) y² + 5y – 24

Solution:

= y² + 8y – 3y – 24

= y (y + 8) – 3 (y + 8)

= (y + 8) (y – 3)

 

(iii) 7y² + 49y +84

Solution:

= 7 (y² + 7y + 12)

= 7 (y² + 4y + 3y + 12)

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

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

 

(iv) 40 + 3x – x²

Solution:

= – [x² – 3x – 40]

= 3 – [x² – 8x + 5x – 40]

= – [x (x – 8) + 5 (x – 8)]

= [(x – 8) (x + 5)]

= (8 – x) (x + 5)

 

(v) m² + 17mn – 84n²

Solution:

= m² + 21mn – 4mn – 84n²

= m (m + 21n) – 4n (m + 21n)

= (m + 21m) (m – 4n)

 

(vi) 117p² + 2pq – 24q²

Solution:

= 117p² + 54pq – 52pq – 24q²

= 9p (13p + 6q) – 3q (13p + 6q)

= (9p – 3q) (13p + 6q)

 

(vii) 15x² – r – 28

Solution:

= 15x² – 21x + 20x – 28

= 3x (5x – 7) + 4 (5x – 7)

= (3x + 4) (5x – 7)

 

(viii) 2x² – x – 21

Solution:

= 2x² – 7x + 6x – 21

= x (2x – 7) + 3 (2x – 7)

= (2x – 7) (x + 3)

 

(ix) 8k² – 22k – 21

Solution:

= 8k² – 28k + 6k – 21

= 4k (2k – 7) + 3(2k – 7)

= (2k – 7) (4k + 3)

 

(x) ^𝟏/_𝟑 x² – 2x – 9

Solution:

= (x²+ 6x−27)/₃

= ¹/₃ [x² – 6x – 27]

= ¹/₃ [x² – 9x + 3x – 27]

= ¹/₃ [x (x – 9) + 3 (x – 9)]

=¹/₃ (x – 9) (x + 3)

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2. Factorize

(i) √𝟓 x² + 2x – 3√𝟓

Solution:

= √5 x2 + 5x – 3x – 3√5

= √5 x (x + √5 ) – 3 ((x +√ 5 )

= (x + √5 ) (√5 x – 3)

 

(ii) √𝟑 a² + 2a – 5√𝟑

Solution:

= √3a² + 5a – 3a – 5√3 – 5√3 – √3

= a (√3 a + 5) – √3 (√3 a + 5)

= (√3 a + 5) (a – √3 )

 

(iii) 7√2 y² – 10y – 4√2

Solution:

= 7√2 y² – 14y + 4y – 4√2

= 7√2 y (y – √2) + 4 (y – 2)

= (y – √2) (7√2 y + 4)

 

(iv) 6√𝟑 z² – 47z – 5√𝟑

Solution:

= 6√3 z² – 45z – 2z – 5√3

= 3√3 z (2z –5√3) – (2z – 5√3)

= (2z – 5√3) (3√3z –1)

 

(v) 4√𝟑x² + 5x – 2√𝟑

Solution:

= 4√3 x² + 8x – 3x – 2√3

= 4x (√3x + 2) – √3(√3x + 2)

= (√3x + 2) (4x –√3)

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3. Factorize

(i) 2 (x + y)² – 9 (x + y) – 5

Solution:

Let x + y = p

= 2p² – 9p – 5

= 2p² – 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

= 2x² – 25x + 12

= 2x² – 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

= 12a² – 25ab + 12b²

= 12a² – 16ab – 9ab + 12b²

= 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)

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4. Factorize

(i) x⁴ – 3x² + 2

Solution:

Put x² = a

a² – 3a + 2

= a² – 2a – a + 2

= a (a – 2) – 1 (a – 2)

= (a – 2) (a – 1)

= (x² – 2) (x² – 1)

= (x – 2 ) (x + 2 ) (x – 1) (x + 1) [∵a² – b² = (a + b) (a – b)]

 

(ii) 4x⁴ + 7x² – 2

Solution:

Put x² = a

= 4a² + 7a – 2

= 4a² + 8a – a – 2

= 4a (a + 2) – 1 (a + 2)

= (a + 2) (4a – 1) [∵a² – b² = (a + b) (a – b)]

= (x² + 2) (4x² – 1)

= (x² + 2) (2x + 1) (2x – 1)

 

(iii) 3x³ – x² – 10x

Solution:

= x (3x² – x – 10)

= x (3x² – 6x + 5x – 10)

= x [3x (x – 2) + 5 (x – 2)]

= x (x – 2) (3x + 5)

 

(iv) 8x³ + 2x²y – 15xy²

Solution:

= x (8x² – 2xy – 15y²)

= x (8x² – 12xy + 10xy – 15y²)

= x [4x (2x – 3y) + 5y (2x – 3y)]

= x (2x – 3y) (4x + 5y)

 

(v) x⁶ – 7x³ – 8

Solution:

Put x³ = a

a³ – 7a – 8 [∵a³ + b³ = (a + b) (a² – ab + b²)]

= a³ – 8a + a – 8 [∵a³ – b³ = (a – b) (a² – ab + b²)]

= a (a – 8) + 1 (a – 8)

= (a – 8) (a + 1)

= (x³ – 3) (x³ + 1)

= (x³ – 3) (x³ + 1)

= (x + 1) (x² – x + 1) (x – 2) (x² + 2x + 4)

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