To solve this equation, we will expand both terms:
Expand (x-3)^6 using the binomial theorem(x-3)^6 = C(6,0)x^6 + C(6,1)x^5(-3) + C(6,2)x^4(-3)^2 + C(6,3)x^3(-3)^3 + C(6,4)x^2(-3)^4 + C(6,5)x(-3)^5 + C(6,6)(-3)^(x-3)^6 = x^6 - 18x^5 + 135x^4 - 540x^3 + 1215x^2 - 1458x + 729
Expand (x^2-2x-1)^3 using the binomial theorem(x^2-2x-1)^3 = C(3,0)x^6 + C(3,1)x^4(-2x) + C(3,2)x^2(-1)^2 + C(3,3)(-1)^(x^2-2x-1)^3 = x^6 - 6x^5 + 12x^4 + 12x^3 - 12x^2 - 6x - 1
Now, substitute these expanded forms back into the original equation:
x^6 - 18x^5 + 135x^4 - 540x^3 + 1215x^2 - 1458x + 729 + x^6 - 6x^5 + 12x^4 + 12x^3 - 12x^2 - 6x - 1 = 0
Combining like terms2x^6 - 24x^5 + 147x^4 - 528x^3 + 1203x^2 - 1464x + 728 = 0
This is a sixth degree polynomial equation. Solving this equation may require the use of numerical methods or graphing techniques.
To solve this equation, we will expand both terms:
Expand (x-3)^6 using the binomial theorem
(x-3)^6 = C(6,0)x^6 + C(6,1)x^5(-3) + C(6,2)x^4(-3)^2 + C(6,3)x^3(-3)^3 + C(6,4)x^2(-3)^4 + C(6,5)x(-3)^5 + C(6,6)(-3)^
(x-3)^6 = x^6 - 18x^5 + 135x^4 - 540x^3 + 1215x^2 - 1458x + 729
Expand (x^2-2x-1)^3 using the binomial theorem
(x^2-2x-1)^3 = C(3,0)x^6 + C(3,1)x^4(-2x) + C(3,2)x^2(-1)^2 + C(3,3)(-1)^
(x^2-2x-1)^3 = x^6 - 6x^5 + 12x^4 + 12x^3 - 12x^2 - 6x - 1
Now, substitute these expanded forms back into the original equation:
x^6 - 18x^5 + 135x^4 - 540x^3 + 1215x^2 - 1458x + 729 + x^6 - 6x^5 + 12x^4 + 12x^3 - 12x^2 - 6x - 1 = 0
Combining like terms
2x^6 - 24x^5 + 147x^4 - 528x^3 + 1203x^2 - 1464x + 728 = 0
This is a sixth degree polynomial equation. Solving this equation may require the use of numerical methods or graphing techniques.