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What's a Prime Polynomial?

Definition: Prime Polynomial

Summary

  1. A prime polynomial is factored as much as it can be
  2. 2x2+14x+3 cannot be factored any further, so it is PRIME
  3. 'x' is also a prime polynomial, because its only factors are 1 and x
  4. A polynomial that is NOT prime is COMPOSITE
  5. 16x2+14x-30 CAN be factored, so it is COMPOSITE
  6. A greatest common factor of 2 can be factored out of 16x2+14x-30
  7. 8x2+7x-15 can be factored into x-1 and 8x+15, so it is also COMPOSITE

Notes

    1. Remember, a prime number is a number whose only factors are 1 and itself
    2. So a prime polynomial is the same thing, except it's a polynomial instead of a number
    1. 2x2+14x+3 is an example of a prime polynomial
    1. This means that we can't factor this polynomial any more than it already is
    1. Since it's prime, it's already factored as much as it can be
    1. There are not two binomials we can multiply together to give us 2x2+14x+3
    1. This is a simple example, but it's also a prime polynomial
    2. 'x' cannot be factored any further
    1. It's the same as with numbers - if it's not prime, then it's composite
    2. So if a polynomial CAN be factored, it is COMPOSITE
    1. 16x2+14x-30 is an example of a composite polynomial
    1. 16x2+14x-30 CAN be factored, so it is composite
    1. Each term in the polynomial has a common factor of 2
    1. 16x2/2=8x
    2. 14x/2=7x
    3. -30/2=-15
    1. Can we factor the polynomial any further?
    2. We CAN factor it further, so it is NOT prime!
    3. It's COMPOSITE
    1. The factors are the things being multiplied together
    2. None of our remaining factors can be factored any further
    1. Once a composite polynomial is factored completely, all its factors should be prime