Question

Factorise (i) a^4 − b^4 (ii) p^4 − 81 (iii) x^4 − (y + z)^4 (iv) x^4 − (x − z)^4 (v) a^4 − 2a^2b^2 + b^4

Answer 1

Solution:

$1) {a}^{4} - {b}^{4} \\ \\ = > ( { {a}^{2} })^{2} - { {(b}^{2} })^{2} \\ \\ = > ( {a}^{2} + {b}^{2} )( {a}^{2} - {b}^{2} ) \\ \\ = > {a}^{4} - {b}^{4} = ( {a}^{2} + {b}^{2} )(a + b)(a - b) \\ \\ formula \: used \: {x}^{2} - {y}^{2} = (x + y)(x - y)$
$2) \: {p}^{4} - 81 \\ \\ = > {p}^{4} - {3}^{4} \\ \\ = > { {(p}^{2} )}^{2} - { {(3}^{2} )}^{2} \\ \\ = > ({p}^{2} - {3}^{2} )({p}^{2} + {3}^{2} ) \\ \\ = > ({p}^{2} + {3}^{2} )(p + 3)(p - 3) \\ \\ {p}^{4} - 81 = ( {p}^{2} + 9)(p + 3)(p - 3)$
$3) {x}^{4} - {(y + z)}^{4} \\ \\ = > ({ {x}^{2} })^{2} - ( { {(y + z)}^{2} })^{2} \\ \\ = > ( {x}^{2} + ( {y + z)}^{2} )( {x}^{2} - ( {y + z)}^{2} )) \\ \\ = > ( {x}^{2} + ( {y + z)}^{2} )(x - y - z)(x + y + z)$
$3) {x}^{4} - {(x - z)}^{4} \\ \\ = > ({ {x}^{2} })^{2} - ( { {(x - z)}^{2} })^{2} \\ \\ = > ( {x}^{2} + ( {x - z)}^{2} )( {x}^{2} - ( {x - z)}^{2} )) \\ \\ = > ( {x}^{2} + ( {x - z)}^{2} )(x - x + z)(x + x - z) \\ \\ = > ( {x}^{2} + {x}^{2} + {z}^{2} - 2xz)(z)(2x - z) \\ \\ {x}^{4} - {(x - z)}^{4}= > z(2x - z)(2 {x}^{2} + {z}^{2} - 2xz)$
$5) \: \: {a}^{4} - 2 {a}^{2} {b}^{2} + {b}^{4} \\ \\ = {a}^{4} - {a}^{2} {b}^{2} - {a}^{2} {b}^{2} + {b}^{4} \\ \\ = > {a}^{2} ( {a}^{2} - {b}^{2} ) - {b}^{2} ( {a}^{2} - {b}^{2} ) \\ \\ = > ( {a}^{2} - {b}^{2} )( {a}^{2} - {b}^{2} ) \\ \\ = > (a + b)(a - b)(a + b)(a - b) \\ \\ = {(a + b)}^{2} ( {a - b})^{2}$

Answer 2

Answer with Step-by-step explanation:

(i) a⁴ − b⁴  

= (a²)² − (b²)²

= (a² − b²) (a² + b²)

By using the formula [(a² − b²) = (a − b) (a + b)]

= (a − b) (a + b) (a² + b²)

 

(ii) p⁴ − 81

= (p²)² -  (9)²

= (p² − 9) (p² + 9)

By using the formula [(a² − b²) = (a − b) (a + b)]

= [(p)² − (3)²] (p² + 9)

= (p − 3) (p + 3) (p² + 9)

 

(iii)x⁴ − (y + z)⁴  

= (x²)² − [(y +z)²]²

= [x² − (y + z)²] [x² + (y + z)²]

By using the formula [(a² − b²) = (a − b) (a + b)]

= [x − (y + z)][ x + (y + z)] [x² + (y + z)²]

= (x − y − z) (x + y + z) [x² + (y + z)²]

 

(iv)x⁴ − (x − z)⁴  

= (x²)² − [(x − z)²]²

= [x² − (x − z)²] [x² + (x − z)²]

By using the formula [(a² − b²) = (a − b) (a + b)]

= [x − (x − z)] [x + (x − z)] [x² + (x − z)²]

= z(2x − z) [x² + x² − 2xz + z²]

= z(2x − z) (2x² − 2xz + z²)

 

(v)a⁴ − 2a²b² + b⁴

= (a²)² − 2 (a²) (b²) + (b²)²

= (a²− b²)²

= [(a − b) (a + b)]²

[(a² − b²) = (a − b) (a + b)]

= (a − b)² (a + b)²

HOPE THIS ANSWER WILL HELP YOU....

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