document.write( "Question 635770: The question states: factor completely or state the polynomial is prime...\r
\n" ); document.write( "\n" ); document.write( "9x^2+25\r
\n" ); document.write( "\n" ); document.write( "and the answer says it is prime... BUT I HAVE NO CLUE WHY :( \r
\n" ); document.write( "\n" ); document.write( "Could you please help me :) God bless your evening :) \n" ); document.write( "
Algebra.Com's Answer #400543 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"9x%5E2%2B25\"
\n" ); document.write( "You conclude that a polynomial is prime after you have tried all the various factoring techniques you have learned and found that the only factors are 1 and the polynomial itself.

\n" ); document.write( "The factorsing techniques you have probably learned are:
    Greatest Common Factor (GCF)
  • Factoring with patterns:
    • \"a%5E2-b%5E2=%28a%2Bb%29%28a-b%29\" (Difference of squares)
    • \"a%5E3%2Bb%5E3=%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29\" (Sum of cubes)
    • \"a%5E3-b%5E3=%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29\" (Difference of cubes)
    • \"a%5E2%2B2ab%2Bb%5E2+=+%28a%2Bb%29%28a%2Bb%29+=+%28a%2Bb%29%5E2\"
    • \"a%5E2-2ab%2Bb%5E2+=+%28a-b%29%28a-b%29+=+%28a-b%29%5E2\"
  • Trinomial factoring: ax^2+bx+c
  • Factoring by grouping
  • Factoring by trial and error of the possible rational roots
All five of these must be attempted and you must fail to find any factors other than 1 and the polynomial itself before you declare a polynomial prime. Let's go through these one-by-one.

\n" ); document.write( "GCF.
\n" ); document.write( "9x^2 = 1 * 3 * 3 * x * x
\n" ); document.write( "25 = 1 * 5 * 3
\n" ); document.write( "The GCF is just 1.

\n" ); document.write( "Patterns.
\n" ); document.write( "Our expression has only two terms. So the only patterns we might be able to use are the one with two terms, the first three. Our expression could be described as a sum of squares: \"%283x%29%5E2%2B%285%29%5E2\". We have patterns for difference of squares, sum of cubes and difference of cubes. Notably missing is a sum of squares. Our expression will not factor using a pattern.

\n" ); document.write( "Trinomial factoring.
\n" ); document.write( "With only two terms we do not have enough terms to factor with this method.

\n" ); document.write( "Factoring by grouping.
\n" ); document.write( "This method requires an even number of terms with at least 4 terms. With only two terms we do not have enough terms to factor with this method.

\n" ); document.write( "Factoring by trial and error of possible rational roots.
\n" ); document.write( "This method is only useful for polynomials of degree 3 or higher. Our polynomial is of degree 2.

\n" ); document.write( "So after all this we found only one factor: 1. So this polynomial is prime. \n" ); document.write( "
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