a = \dfrac{\sqrt{101} +1 }{2}

Then

2a = \sqrt{101}+1

Rearrange the terms to make the term with square root on one side

2a-1 = \sqrt{101}

Square both sides

4a^2-4a+1 = 101

Then

4a^2-4a = 100

a^2 = a+25
(1)

Now let's reduce the degree of polynomial using the result

(a^3-26a-23)^7

=[a(a+25)-26a-23]^7

=(a^2+25a-26a-23)^7

=(a^2-a-23)^7

=(a+25-a-23)^7

=2^7

=128

Steven Zheng posted 20 hours ago

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