1.

Factorise : `(a) x^(4)+4x^(2)+16 " " (ii) x^(4)+4`

Answer» `(a) " We have" , x^(4)+4x^(2)+16=(x^(2))^(2)+2(x^(2))(4)+(4)^(2)-8x^(2)+4x^(2)`
[we are trying to make the formula `a^(2)+2ab+b^(2)=(a+b)^(2)]`
`=(x^(2)+4)^(2)-4x^(2)=(x^(2)+4)^(2)-(2x)^(2)`
`=(x^(2)+4+2x)(x^(2)+4-2x) " " [because a^(2)-b^(2)=(a+b)(a-b)]`
`=(x^(2)+2x+4)(x^(2)-2x+4)`
`(b) x^(4)+4=(x^(2))^(2)+(2)^(2)`
`=ubrace((x^(2))^(2)+2(x^(2))(2)+(2)^(2))-2(x^(2))(2)` (adding and subtracting the same quantity)
`(x^(2)+2)^(2)-4x^(2)`
`=(x^(2)+2)^(2)-(2x)^(2)=(x^(2)+2+2x)(x^(2)+2-2x)=(x^(2)+2x+2)(x^(2)-2x+2)`


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