%5E3-(x%2B1)(x%5E2-x%2B1)-(3x%2B1)(1-3x).jpeg "(x-1)^3-(x+1)(x^2-x+1)-(3x+1)(1-3x)")
Simplifying the Expression (x-1)^3-(x+1)(x^2-x+1)-(3x+1)(1-3x)
This article will guide you through the process of simplifying the given algebraic expression:
(x-1)^3-(x+1)(x^2-x+1)-(3x+1)(1-3x)
Let's break down the simplification step-by-step:
Expanding the Expressions
-
(x-1)^3: We can use the binomial theorem or simply expand it directly: (x-1)^3 = (x-1)(x-1)(x-1) = (x^2 - 2x + 1)(x-1) = x^3 - 3x^2 + 3x - 1
-
(x+1)(x^2-x+1): This is a special product known as the "sum of cubes" pattern: (x+1)(x^2-x+1) = x^3 + 1
-
(3x+1)(1-3x): This is a simple product that can be expanded using the distributive property: (3x+1)(1-3x) = 3x - 9x^2 + 1 - 3x = -9x^2 + 1
Combining the Terms
Now, substitute the expanded expressions back into the original equation:
(x-1)^3-(x+1)(x^2-x+1)-(3x+1)(1-3x) = (x^3 - 3x^2 + 3x - 1) - (x^3 + 1) - (-9x^2 + 1)
Simplifying the Equation
Finally, combine like terms and simplify:
x^3 - 3x^2 + 3x - 1 - x^3 - 1 + 9x^2 - 1 = 6x^2 + 3x - 3
Therefore, the simplified form of the given expression is 6x^2 + 3x - 3.