If x=(√3+1)/2, find the value of 4x^3+2x^2-8x+7?

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If x=(√3+1)/2, find the value of 4x^3+2x^2-8x+7?

Given x = (√3 +1)/2

4((√3 +1)/2)^3 + 2((√3 +1)/2)^2 − 8((√3 +1)/2) + 7

Now, 4(√3 +1)^3/8 + 2(√3 +1)^2/4 − 8(√3 +1)/2 + 7

=> (√3 +1)^3/2 + (√3 +1)^2/2 − 4(√3 +1) + 7

We know that,(a + b)^n = ∑[k=0,n] C(n,k) * a^(n−k) * b^k

Hence (a + b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3

and (a + b)^2 = 1a^2 + 2ab + 1b^2

=> (3√3 + 9 + 3√3 +1)/2 + (3 + 2√3 +1)/2 − 4(√3 +1) + 7

=> (6√3 + 10)/2 + (2√3 +4)/2 − 4(√3 +1) + 7

=> 3√3 + 5 + √3 + 2 − 4√3 − 4 + 7 = 10

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