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Find (4x − 2y + 3z) 2.
We know that
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac
∴ (4x − 2y + 3z)2
= (4x)2+ (−2y)2 + (3z)2 + 2 × 4x × −2y + 2 × −2y × 3z + 2 × 4x × 3z
=16x2 + 4y2 +9z2 - 16xy -12yz + 24xz
Factorise 32x4 - 2y4.
32x4 - 2y4 = 2(16x4 − y4)
= 2[(4x2)2 - (y2)2]
= 2(4x2 − y2)(4x2 + y2)… using a2 − b2 = (a-b) (a + b)
= 2(2x - y) (2x + y)(4x2 + y2) … using a2 − b2 = (a - b) (a + b)
Simplify (4a − 2b + 3c) 2 - (3a − 5b − 2c) 2.
(a + b + c)2 = a2 + b2 + c2+ 2ab + 2bc + 2ac
∴ (4a − 2b + 3c)2 = 16a2 + 4b2 + 9c2 - 16ab − 12bc + 24ac
and (3a − 5b − 2c)2 = 9a2 + 25b2 + 4c2 - 30ab + 20bc - 12ac
∴ (4a − 2b + 3c)2 - (3a − 5b − 2c)2
= 16a2 + 4b2 + 9c2 - 16ab − 12bc + 24ac − 9a2 − 25b2 − 4c2 + 30ab − 20bc + 12ac
= 7a2 − 21b2 + 5c2 + 14ab − 32bc + 36ac
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