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Chapter 1: Problem 27

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

Short Answer

Expert verified
The zeros of the function \( \frac{1}{2} x^{3}-x \) are \( x = 0, x = \sqrt{2}, x = -\sqrt{2} \).

Step by step solution

01

Setting the Equation to Zero

To find the zeros of the function themust be first set to zero. Which gives \(\frac{1}{2} x^{3}-x = 0\).
02

Factor Out Common Factor

The next step is simplifying the equation by factoring out the common factor, which is \( x \). This simplification leads to the equation \( x(\frac{1}{2} x^{2}-1) = 0\).
03

Set each Factor Equal to Zero

In order to find potential solutions, each factor from the previous equation should be set equal to zero: (1) \( x = 0\) and (2) \(\frac{1}{2} x^{2}-1 = 0\).
04

Solve the Equations

The first equation \( x = 0 \) already shows a solution: \( x = 0 \). The second equation can be solved like this: \(\frac{1}{2} x^{2}=1\), then \( x^{2} = 2 \), and finally \( x = \sqrt{2} \) or \( x = -\sqrt{2} \).

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