Question

Find all real values of \(x\) such that \(f(x)=0\) . \(f(x)=x^{3}-x\)

Short answer

The real solutions for the given equation are \(x = -1\), \(x = 0\), and \(x = 1\).

Step-by-step solution

Identify the Polynomial Expression

In the problem, the given function is \(f(x) = x^{3} - x\). We need to find the real values of \(x\) when \(f(x) = 0\). Thus, we have the equation \(x^{3} - x = 0\).

Factorize the Polynomial Expression

The given polynomial \(x^{3} - x\) can be factorized using common factors. The term \(x\) is common in both \(x^{3}\) and \(x\). Thus, factorizing the polynomial gives us \(x(x^{2} - 1) = 0\).

Solve Each Factor for Zero

In order to find the roots of the equation, set each factor of the equation equal to zero. Therefore, we have \(x = 0\) and \(x^{2} - 1 = 0\). Solving \(x^{2} - 1 = 0\) we have two more roots, which are \(x = -1\) and \(x = 1\).

Key concepts

Polynomial Equations

Polynomial equations are mathematical expressions that involve variables raised to whole number powers. These equations can have one or more terms, generally separated by addition or subtraction.
Polynomials are usually written in standard form where terms are arranged in descending order of their degree. For example, the polynomial \(x^3 - x\) is a cubic polynomial because the highest exponent is 3.
To solve polynomial equations like \(f(x) = 0\), we seek to find values of \(x\) that make the polynomial equal to zero. These values are called roots or zeros of the polynomial. Understanding the degree of the polynomial is crucial, as it tells us the maximum number of roots the polynomial can have.
For instance, a cubic polynomial can have up to three real roots. Finding these roots helps us understand the behavior and points of intersection of polynomial graphs with the x-axis.

Factorization

Factorization is the process of breaking down a polynomial into simpler 'factors' that, when multiplied together, give back the original polynomial. This method is useful to solve polynomial equations because it allows us to set each factor to zero to find the equation's roots.
Consider the polynomial \(x^3 - x\). By taking out the greatest common factor, which here is \(x\), we can rewrite it as \(x(x^2 - 1) = 0\). This factorization simplifies our task of finding the roots, as it divides the problem into smaller parts.
The quadratic part, \(x^2 -1\), can further be factorized using the identity \(a^2 - b^2 = (a + b)(a - b)\), which gives \((x - 1)(x + 1)\).
Thus, by solving \(x(x-1)(x+1) = 0\), the solutions appear as \(x = 0, x = 1, \text{and } x = -1\). Factorization is key in breaking down complex polynomials to find solutions easily.

Cubic Functions

Cubic functions are polynomial functions where the highest power of the variable is 3. These functions are of the form \(g(x) = ax^3 + bx^2 + cx + d\). Cubic functions can exhibit various shapes in their graphs, which can include one inflection point and up to three real roots.
The cubic function \(x^3 - x\) is a special form where the coefficients of some terms are zero—meaning the function has no \(x^2\) or constant term. This simplicity allows it to be solved relatively easily by factorization.
Graphically, the behavior of cubic functions can be complex. They can rise and fall multiple times, and intersect the x-axis at points found by solving \(f(x) = 0\). These points of intersection are the roots. In the case of \(x^3 - x = 0\), the graph would intersect the x-axis at \(x = 0, x = 1, \text{and } x = -1\). Cubic functions provide a diverse range of applications due to their peculiarity and wide array of shapes, making them essential in the study of polynomial functions.