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A polynomial g(x) satisfies the functional equation g(x) * g(y) = g(x) + g(y) + g(x*y) - 2 for all real numbers x and y. Given that g(2) = 5, find the value of g(3).

1. 10
2. 12
3. 16
4. 26

Correct answer: 10

Solution

Substitute x = y = 1: g(1)² = g(1) + g(1) + g(1) - 2 = 3*g(1) - 2, so g(1)² - 3*g(1) + 2 = 0, giving (g(1)-1)*(g(1)-2) = 0, so g(1) = 1 or g(1) = 2. Try g(x) = xⁿ + 1: g(x)*g(y) = (xⁿ+1)*(yⁿ+1) = xⁿ*yⁿ + xⁿ + yⁿ + 1. RHS: g(x)+g(y)+g(xy)-2 = (xⁿ+1)+(yⁿ+1)+((xy)ⁿ+1)-2 = xⁿ+yⁿ+(xy)ⁿ+1. These match! So g(x) = xⁿ + 1. Now g(2) = 2ⁿ + 1 = 5 gives 2ⁿ = 4, so n = 2. Therefore g(x) = x² + 1 and g(3) = 9 + 1 = 10.

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