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Chapter 1: Problem 19

Find the zeros of the function algebraically. $$f(x)=\frac{1}{2} x^{3}-x$$

Short Answer

Expert verified
The zeros for the function are \(x=0\), \(x=\sqrt{2}\), and \(x=-\sqrt{2}\).

Step by step solution

01

Set the function equal to zero

To find the zeros of the function \(f(x)\), the function is set equal to zero. This gives the equation: \[0 = \frac{1}{2} x^{3}-x\]
02

Simplify the equation

For simplification purposes, multiply every term in the equation by 2 to eliminate the fraction: \[0 = x^{3} - 2x\]
03

Factoring

Next step involves factoring the equation. In this case, we can factor out an \(x\) from each term: \[0 = x(x^{2} - 2)\]
04

Solve for x's

Finally, set each factor equal to zero and solve for \(x\): \[\begin{align*} x & = 0 \ x^{2} - 2 & = 0 \end{align*}\] The second equation gives us two more zeros after taking the square root of each side: \[\begin{align*} x^{2} & = 2 \ x & = \pm \sqrt{2} \end{align*}\]

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