factorise a^4 −(b+c)^4


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Question Number 27449 by rohith siddharth last updated on 07/Jan/18

$f a c t o r i s e a^{4} - {(b + c)}^{4}$
	Answered by $@ty@m last updated on 07/Jan/18
	$Let b+c=d a^4 −d^4 =(a^2 )^2 −(d^2 )^2 =(a^2 −d^2 )(a^2 +d^2 ) =(a+d)(a−d)(a^2 +d^2 ) =(a+b+c)(a−b−c){a^2 +(b+c)^2 }$
	$L e t b + c = d$ $a^{4} - d^{4}$ $= {(a^{2})}^{2} - {(d^{2})}^{2}$ $= (a^{2} - d^{2}) (a^{2} + d^{2})$ $= (a + d) (a - d) (a^{2} + d^{2})$ $= (a + b + c) (a - b - c) {a^{2} + {(b + c)}^{2}}$
	Commented by rohith siddharth last updated on 07/Jan/18

	$I want answer step by step$
	Commented by $@ty@m last updated on 07/Jan/18

	$I s i t o k a y n o w ?$
	Commented by rohith siddharth last updated on 07/Jan/18

	$ok thank you$
	Answered by Joel578 last updated on 07/Jan/18

	$Recall that p^{2} - q^{2} = (p + q) (p - q)$ $a^{4} - {(b + c)}^{4} = {(a^{2})}^{2} - {({(b + c)}^{2})}^{2}$ $= (a^{2} - {(b + c)}^{2}) (a^{2} + {(b + c)}^{2})$ $= (a + b + c) (a - (b + c)) (a^{2} + {(b + c)}^{2})$ $= (a + b + c) (a - b - c) (a^{2} + {(b + c)}^{2})$ $R e call also p^{2} + q^{2} = {(p + q)}^{2} - 2 p q$ $= (a + b + c) (a - b - c) (a^{2} + {(b + c)}^{2})$ $= (a + b + c) (a - b - c) [{(a + b + c)}^{2} - 2 a (b + c)]$
	Commented by rohith siddharth last updated on 07/Jan/18

	$thank you bro$
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