If x = 1 + √2 + √3, find x4 – 4x3 – 4x2 + 16x + 8.
Explanation:
x = 1 + √2 + √3
⇒ x - 1 = √2 + √3
Squaring both sides, we get
⇒ x2 – 2x + 1 = 2 + 3 + 2√6
⇒ x2 – 2x - 4 = 2√6
Again, squaring both sides, we get
⇒ x4 + 4x2 + 16 – 4x3 – 8x2 + 16x = 24.
⇒ x4 – 4x3 - 4x2 + 16x + 16 = 24.
⇒ x4 – 4x3 - 4x2 + 16x = 8.
Now, we have
x4 – 4x3 – 4x2 + 16x + 8 = 8 + 8 = 16.
Hence, 16.
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