Find all real solutions to the equation: x^(5)-5x^(4)+10x^(3)-10x^(2)+5x-1=0

Pattern Recognition: Notice the pattern in the coefficients, it looks like the binomial theorem expansion of (x-1)^5. Let's check if (x-1)^5 equals the given polynomial. (x-1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 Yep, it matches exactly.Equation Simplification: Since (x-1)^5 equals the given polynomial, we can write the equation as: (x-1)^5 = 0Finding Real Solutions: To find the real solutions, we need to solve for x in the equation (x-1)^5 = 0. Taking the fifth root of both sides gives us x - 1 = 0.Final Solution: Now, solve for x by adding 1 to both sides. x = 1

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Find all real solutions to the equation: $\newline$ $x^{5}-5x^{4}+10x^{3}-10x^{2}+5x-1=0$

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Q. Find all real solutions to the equation:

\newline

x^{5}-5x^{4}+10x^{3}-10x^{2}+5x-1=0

Pattern Recognition: Notice the pattern in the coefficients, it looks like the binomial theorem expansion of $(x-1)^5$ . Let's check if $(x-1)^5$ equals the given polynomial. $(x-1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1$ Yep, it matches exactly.
Equation Simplification: Since $(x-1)^5$ equals the given polynomial, we can write the equation as: $\newline$ $(x-1)^5 = 0$
Finding Real Solutions: To find the real solutions, we need to solve for $x$ in the equation $(x-1)^5 = 0$ . Taking the fifth root of both sides gives us $x - 1 = 0$ .
Final Solution: Now, solve for $x$ by adding $1$ to both sides. $\newline$ $x = 1$