FACTORIZATION – EXERCISE 3.2.4 – Class 9

1. Resolve into factors

(i) x⁴ + y⁴ – 7x² y²

Solution:

= x⁴ + y⁴ – 2x² y² – 2x² y² – 7x² y²

= x⁴ + y⁴ + 2x² y² – 9x² y²

= (x² + y²)² – (3xy)²

= (x² + y² + 3xy) (x² + y² – 3xy)

 

(ii) 4x⁴ + 25y⁴ + 10x²y²

Solution:

Take a = √2x ;b = √5y

a⁴ + b⁴ + a² b² = (a² + b² + ab) (a² + b² – ab)

4x⁴ + 25y⁴ + 10x²y² = (√2 x)⁴ + (√5 y)⁴ + 2 x² 5 y²

= [( 2 x²) + ( 5 y²) + (√10 xy] [( 2 x²) + ( 5 y²) – (√10 xy]

= (2x² +5y² + 10 xy) (2x² +5y² – 10 xy)

(iii)9a⁴+100b⁴+30a²b²

Solution:

Take a = √3a ;b = √10b

a⁴ + b⁴ + a² b² = (a² + b² + ab) (a² + b² – ab)

9a⁴+100b⁴+30a²b² = (√3a)⁴+(√10b)⁴+3a².10b²

= (3a²+10b²+30a²b²) (3a²+10b²– 30a²b²)

(iv)81a4 +9a2b2+b4

Solution:

Take a = 3a ;b = b

a⁴ + b⁴ + a² b² = (a² + b² + ab) (a² + b² – ab)

81a⁴ +9a²b²+b⁴ = (3a)⁴+(3a)²(b)²+b⁴

= (9a²+b²+3ab)(9a²+b²-3ab)

 

(v) x⁴ – 6x²y² + y⁴

= x⁴ + y⁴ + 2x²y² – 2x²y² – 6x²y²

= (x² + y²)² – ( 8 xy)²

= (x² + y² + 8 xy) (x² + y² – 8 xy)

 

1. vi) m⁴ + n⁴ – 18m²n²

= m⁴ + n⁴ + 2m²n² – 2m²n² – 18m²n²

= (m² + n²)² – 20 m²n²

= (m² + n² + √20 mn) (m² + n² – √20 mn)

 

(vii) 4m⁴ + 9n⁴ – 24m²n²

= 4m⁴ + 9n⁴ + 12 m²n² – 12 m²n² – 24m²n²

= (2m² + 3n²)² – 36mn

= (2m² + 3n² + 6mn) (2m² + 3n² – 6mn)

 

(viii) 9x⁴ + 4y⁴ + 11x²y²

= 9x⁴ + 4y⁴ + 12x²y² – 12x²y² + 11x²y²

= (3x²)² + (2y²)² + 2 (3x)² (2y)² – x²y²

= (3x² + 2y²)² – x²y²

= (3x² + 2y² + xy) (3x² + 2y² – xy)

---

2. Find the factors of the following

(i) x⁴ + 9x²+81

Solution:

= x⁴+9x²+81+9x²-9x²

= (x²)²+18x²+81-9a²

=(x²+9)²-(3a)²

=( x²+9 +3a)( x²+9 – 3a)

 

(ii) a⁴ + 4a² + 16

Solution:

= a⁴ + 4a² + 16 + 4a² – 4a²

= (a²)² + 8a² + (4)² – 4a²

= (a² + 4)² – (2a)²

= (a² + 4 – 2a) (a² + 4 + 2a)

---

3. Factorise the following

(i) 64a⁴ + 1

Solution:

Adding and subtracting 16a² we get

64a4 + 1 + 16a² – 16a²

= (8a²)² + 1 + 16a² – 16a²

= (8a² + 1)² – (4a)²

= (8a² + 1 + 4a) (8a² + 1 – 4a)

 

(ii) 3x⁴ + 12y⁴

Solution:

3(x⁴ + 4y⁴)

3[(x²)² + (²y²)²]

By adding and subtracting 2ab i. e.

2 x x² x 2y = 4x²y²

= 3 [(x²)² + 2(y²)² + 4x²y² – 4x²y²]

= 3 [(x² + 2y²)² – (2xy)²]

= 3 [(x² + 2y2 + 2xy) (x² + 2y² – 2xy)]

 

(iii) 4x⁴ + 81y⁴

Solution:

(2x²)² + (9y²)²

By adding and subtracting 2ab i. e.

2 x x² x 9y² = 36x²y²

= (2x²)² + (9y²)² + 36x²y² – 36x²y²

= (2x² + 9y²)² – (6xy)²

= (2x² + 9y² + 6xy) (2x² + 9y² – 6xy)

 

(iv) a⁸ – 16b⁸

Solution:

(a⁴)² – (4b⁴)²

Using a² – b² = (a + b) (a – b)

= (a⁴ + 4b⁴) (a⁴ – 4b⁴)

= (a⁴ + 4b⁴) [(a²)² – (2b²)²]

= (a⁴ + 4b⁴) (a² + 2b²) (a² – 2b²)

---