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Prime Polynomial
A prime polynomial is a polynomial that cannot be written as a product of two or more simpler polynomial factors — it is already in its simplest factored form. Just as a prime number has no factors other than and itself, a prime polynomial has no polynomial factors other than and itself. Not every polynomial can be factored, so when a factoring strategy (such as looking for a GCF, factoring by grouping, or searching for factor pairs of the constant term) fails to produce factors, the polynomial may be prime.
Some trinomials are prime. The only way to be certain that a trinomial is prime is to list all possible factor pairs of the constant term, check whether any pair has the required sum equal to the middle coefficient, and show that none of them work. If no factor pair produces the correct sum, the trinomial cannot be factored and is therefore prime.
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