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Chapter 1: Problem 43

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

Short Answer

Expert verified
The real values of \(x\) such that \(f(x)=0\) are \(x = -1, 0, 1\)

Step by step solution

01

Set up the equation

Set the function \(f(x)\) equal to zero, and then solve for \(x\): \[ f(x) = x^{3}-x = 0 \]
02

Factor the polynomial

Notice that every term has an \(x\), so factor out an \(x\): \[ x(x^{2}-1) = 0 \] The expression in the parenthesis is a difference of squares, which can be further factored as: \[ x(x-1)(x+1) = 0 \]
03

Set each factor equal to zero

Now, set each factor of the polynomial equal to zero: \[ x = 0, x - 1 = 0, x + 1 = 0 \]
04

Solve for \(x\)

Solving these equations gives the solutions: \[ x = 0, x = 1, x = -1 \]

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