Question

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Answer 1

Step-by-step explanation:

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Answer 2

Question :-

1. Factorise:

a) (a + b)² - (x - y)²

b) x⁸ - 1

c) (x + 2y)⁴ - 1

d) 49x²y² - z²

2. Factorise the trinomials:

a) x² + 2 + 1/x²

b) 9x² + 4y² + 12xy

c) 9x² + 24x + 16

Solution :-

The identity which is mainly used in this question is:

a² - b² = (a + b) (a - b)

1. Factorise:

a) (a + b)² - (x - y)²

→ This is of the form a² - b² = (a + b) (a - b)

Applying the above identity,

(a + b)² - (x - y)²

= (a + b + x + y) (a + b - (x - y))

= (a + b + x + y) (a + b - x + y)

∴ (a + b)² - (x - y)² = (a + b + x + y) (a + b - x + y)

b) x⁸ - 1

→ This can also be written as:

x⁸ - 1⁸

= (x⁴)² - (1⁴)²

= (x⁴ + 1⁴) (x⁴ - 1⁴)

= (x⁴ + 1⁴) {(x²)² - (1²)²}

= (x⁴ + 1⁴) (x² + 1²) (x² - 1²)

= (x⁴ + 1) (x² + 1) (x + 1) (x - 1)

∴ x⁸ - 1⁸ = (x⁴ + 1) (x² + 1) (x - 1) (x - 1)

c) (x + 2y)⁴ - 1

→ This can be written as:

(x + 2y)⁴ - 1⁴

= {(x + 2y)²}² - (1²)²

= (x + 2y + 1) (x + 2y - 1)

∴ (x + 2y)⁴ - 1 = (x + 2y + 1) (x + 2y - 1)

d) 49x²y² - z²

→ This can be written as:

(7xy)² - z²

= (7xy + z) (7xy - z)

∴ 49x²y² - z² = (7xy + z) (7xy - z)

2. Factorise the trinomials:

The main identity used in this question is:

(a + b)² = a² + b² + 2ab

a) x² + 2 + 1/x²

→ This is of the form: a² + 2ab + b²

So, it can be written as (a + b)²

= (x + 1/x)²

Also, when we split (x + 1/x)² we get (x² + 2 + 1/x²) which verifies our answer.

∴ x² + 2 + 1/x² = (x + 1/x)²

b) 9x² + 4y² + 12xy

→ This is also of the form a² + b² + 2ab

where:

• a² = 9x² and so a = 3x
• b² = 4y² and so b = 2y

Now, 9x² + 4y² + 12xy

= (3x + 2y)²

∴ 9x² + 4y² + 12xy = (3x + 2y)²

c) 9x² + 24x + 16

→ This is of the form a² + 2ab + b²

where:

• a² = 9x² and so a = 3x
• b² = 16 and so b = 4

Now, 9x² + 24x + 16

= (3x + 4)²

∴ 9x² + 24x + 16 = (3x + 4)²