What are the Zeros of the Cubic Polynomial (x^3 - x)?
This article explains the process of finding the zeros of the cubic polynomial x3 - x. We will break down the solution step-by-step and provide a clear explanation. By the end, you'll understand how to determine the roots of this polynomial.
Step-by-Step Guide to Finding the Zeros
The polynomial (x^3 - x) can be simplified as follows:
(x^3 - x x(x^2 - 1))
This factorization helps us identify the zeros of the polynomial. We start by solving the equation:
(x^3 - x 0)
This equation can be rewritten as:
(x(x^2 - 1) 0)
Next, we factorize further:
(x(x - 1)(x 1) 0)
A polynomial is zero if and only if one of the factors is zero. Therefore, we set each factor to zero:
(x 0) (x - 1 0 Rightarrow x 1) (x 1 0 Rightarrow x -1)Thus, the zeros of the polynomial (x^3 - x) are:
(x -1) (x 0) (x 1)Explanation of the Zeros
The zeros of a polynomial (p(x)) are the values of (x) that make the polynomial equal to zero. For the polynomial (x^3 - x), the zeros are the solutions to the equation:
(x^3 - x 0)
This can be written as:
(x(x - 1)(x 1) 0)
Setting each factor to zero gives us:
(x 0) (x - 1 0 Rightarrow x 1) (x 1 0 Rightarrow x -1)Hence, the zeros are -1, 0, and 1. These are the points where the graph of the polynomial function (f(x) x^3 - x) intersects the x-axis.
Conclusion
In summary, the zeros of the cubic polynomial (x^3 - x) are -1, 0, and 1. By factoring the polynomial, we can easily identify these points where the polynomial equals zero. This process is fundamental in understanding and analyzing polynomial functions.
Further Reading and Questions
For a deeper understanding, you may want to explore related concepts such as polynomial factorization, the Fundamental Theorem of Algebra, and the behavior of polynomial functions. If you have any questions or need further assistance, feel free to ask in the comments!
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