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Factorize each of the following algebrai...

Factorize each of the following algebraic expressions:
(1) `a^2+2a b+b^2-c^2`
(2) `49-x^2-y^2+2x y`
(3) `a^2+4b^2-4a b-4c^2`
(4) `x^2-y^2-4x z+4z^2`

Text Solution

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(1) `a^2+2a b+b^2-c^2`
Using the identity:
`(a+b)^2=a^2+2ab+b^2`
`=a^2+2a b+b^2-c^2`
`=(a+b)^2-c^2`
`=(a+b+c)(a+b-c)`

(2) `49-x^2-y^2+2x y`
Using the identity:
`(x-y)^2=x^2-2xy+y^2`
`=x^2-2xy+y^2-c^2`
`=49-x^2-y^2+2x y`
`=7^2-(x-y)^2`
`=(7-x+y)(7+x-y)`

(3) `a^2+4b^2-4a b-4c^2`
Using the identity:
`(a-b)^2=a^2-2ab+b^2`
`=a^2+4b^2-4a b-4c^2`
`=a^2-2timesatimes2b+(2b)^2-(2c)^2`
`=(a-2b-2c)(a-2b+2c)`

(4)`x^2-y^2-4x z+4z^2`
Using the identity:
`(x-y)^2=x^2-2xy+y^2`
`=x^2-4xz+4z^2-y^2`
`=x^2-2timesxtimes2z+(2z)^2-y^2`
`=(x-2z)^2-y^2`
`=(x-2z-y)(x-2z+y)`
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