Yes, though you could go into a bit more of an explanation as to why there would have to be a first degree factor.Justabeginner said:Is this approach correct?
With over 300 posts in this forum, you should have learned enough of the ropes here to write exponents clearly.Justabeginner said:Homework Statement
If p(x) ∈F[x] is of degree 3, and p(x)=a0+a1∗x+a2∗x2+a3∗x3, show that p(x) is irreducible over F if there is no element r∈F such that a0+a1∗r+a2∗r2+a3∗r3 =0.
Justabeginner said:Homework Equations
The Attempt at a Solution
Is this approach correct?
If p(x) is reducible, then there exists ax + b such that a, b ε F and a≠0. And p(x) = (ax + b)(cx^2 + dx + e). Then an r exists such that p(r) = 0.
Thank you.
An irreducible polynomial of degree 3 is a third degree polynomial that cannot be factored into lower degree polynomials with coefficients in the same field. This means that it cannot be broken down into simpler terms and has no rational roots.
Studying irreducible polynomials of degree 3 is important because they have many important applications in mathematics and other fields such as engineering and physics. They are also used in cryptography and coding theory.
A polynomial of degree 3 can be determined to be irreducible by using the Rational Root Theorem, which states that if a polynomial with integer coefficients has a rational root, it must be a factor of the constant term. If no rational roots are found, then the polynomial is irreducible.
A reducible polynomial of degree 3 can be factored into lower degree polynomials with coefficients in the same field, while an irreducible polynomial of degree 3 cannot be factored. This means that a reducible polynomial can be broken down into simpler terms, while an irreducible polynomial cannot.
Yes, an irreducible polynomial of degree 3 can have complex roots. This is because the degree of a polynomial refers to the highest power of the variable, not the number of distinct roots. An irreducible polynomial can have either real or complex roots.