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Chapter 3: Problem 70

Explain why the equation \(x^{4}+6 x^{2}+2=0\) has no rational roots.

Short Answer

Expert verified
The analysis and verification of potential rational roots proves that the equation \(x^{4}+6 x^{2}+2=0\) does not have any rational roots.

Step by step solution

01

Identify potential rational roots

Utilizing the Rational Root Theorem, potential rational roots can be defined as \(\pm1, \pm2\).
02

Verification of the roots

Substitute these potential roots into the equation \(x^{4}+6 x^{2}+2=0\) to verify if any of them are indeed roots of the equation.
03

Check \(x = 1\)

For \(x = 1\), the equation becomes \(1+6+2=9\), which is not equal to zero, so \(x = 1\) is not a root.
04

Check \(x = -1\)

For \(x = -1\), the equation becomes \((-1)^4+6(-1)^2+2=9\), which is not equal to zero, so \(x = -1\) is not a root.
05

Check \(x = 2\)

For \(x = 2\), the equation becomes \(16+24+2=42\), which is not equal to zero, so \(x = 2\) is not a root.
06

Check \(x = -2\)

For \(x = -2\), the equation becomes \((-2)^4+6(-2)^2+2=42\), which is not equal to zero, so \(x = -2\) is not a root.
07

Conclusion

After checking all potential rational roots, none of them turned out to be a root of the equation. Hence, it can be concluded that our original equation \(x^{4}+6 x^{2}+2=0\) has no rational roots.

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