# Abstract algebra

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Consider the cubic polynomial:

See attached file for full problem description.

Prove that if all 3 zeros are real, then all 3 coefficients are real.

Consider the cubic polynomial:

P(z) = z3 + a1z2 + a2z + a3

Prove that if all 3 zeros are real, then all 3 coefficients are real.

https://brainmass.com/math/basic-algebra/abstract-algebra-polynomial-proof-64527

#### Solution Preview

Consider the cubic polynomial:

P(z) = z3 + a1z2 + a2z + a3

Prove that if all 3 zeros are real, then all 3 coefficients are real.

Proof

Let r1 ,r2 ,r3 are three real zeros then

P(z) = (z - r1) (z - r2) (z - r3) = 0

comparing the coefficients

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#### Solution Summary

If all three zeros are real, then all 3 coefficients are real.

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