A Nice Polynomial Expression

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[@mystychief]

? = x(x^4-29) makes it easier. x = sqrt(2)+1. x² = 3+2sqrt(2). x^4 = 17+12sqrt(2). Fill in ? = (sqrt(2)+1)(17+12sqrt(2)-29) = (sqrt(2)+1)(-12+12sqrt(2)) = ... = 12.

[@ianfowler9340]

Or you could just use your method on x=√2+1. AKA The Silver Ratio! x^2=3√2+1 x^3=5√2+7 x^5=29√2+41 x^5-29x=41-29=12 But keeping the x as a variable is a sweet solution.

[@nasrullahhusnan2289]

x⁵-29x=x(x⁴-29) x=1+sqrt(2) --> x²=3+2sqrt(2) x⁴=17+12sqrt(2) x⁴-29=-12[1-sqrt(2)] x(x⁴-29)=-12[1+sqrt(2)][1-sqrt(2)] =-12[(-1) =12 Common procedure: x=1+sqrt(2) x⁵=[1+sqrt(2)]⁵ =1+5sqrt(2)+10(2)+10[2sqrt(2)] +5(20)+4sqrt(2) =121+29sqrt(2) 29x=29+29sqrt(2) x⁵-29x=92

[@paulortega5317]

Or a recursive approach... x₁ y₁ x₂ = x₁ + 2y₁ y₂ = x₁ + y₁ Also x₃ = 2x₂ + x₁ and y₃ = 2y₂ + y₁ 1 1 3 2 7 5 17 12 41 29 Plus the value pairs, alternately, are solutions for the diophantine equations x² = 2y² - 1 and x² = 2y² + 1

[@elmer6123]

x^2=2x+1; x^3=2x^2+x=5x+2; x^5=(2x+1)(5x+2)=10x^2+9x+2=29x+12 x^5-29x=29x+12-29x=12

[@trojanleo123]

Answer = 12