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Factorise by completing the squares.
(i) a
4
+ a
2
+ 1
(ii) y
4
+ 5y
2
+ 9
(iii) x
4
+ 4
(iv) x
4
+ 4x
2
+ 3
(i) a
4
+ a
2
+ 1
= a
4
+ a
2
+ 1 + a
2
- a
2
= a
4
+ 2a
2
+ 1 - a
2
= (a
2
)
2
+ 2a
2
+ 1 - a
2
= (a
2
+ 1)
2
- a
2
= (a
2
+ 1 + a) (a
2
+ 1 - a)
= (a
2
+ a + 1) (a
2
- a + 1)
(ii) y
4
+ 5y
2
+ 9
= (y
2
)
2
+ 5y
2
+ (3)
2
+ y
2
- y
2
= (y
2
)
2
+ 6y
2
+ (3)
2
- y
2
= (y
2
+ 3)
2
- y
2
= (y
2
+ y + 3) (y
2
- y + 3)
(iii) x
4
+ 4
= x
4
+ 4 + 4x
2
- 4x
2
= (x
2
)
2
+ (2)
2
+ (2x)
2
- 4x
2
= (x
2
+ 2)
2
- (2x)
2
= (x
2
+ 2x + 2) (x
2
- 2x + 2)
(iv) x
4
+ 4x
2
+ 3
= x
4
+ 4x
2
+ 4 - 1
= x
4
+ 4x
2
+ 3 + 1 - 1
= (x
2
)
2
+ 2 × 2x
2
+ (2)
2
- 1
= (x
2
+ 2)
2
- 1
= (x
2
+ 2 + 1) (x
2
+ 2 - 1) = (x
2
+ 3) (x
2
+ 1)
13
