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Note: I assume that the 3x^2 term should really be 3x^3.
\n" ); document.write( "Note 2: You could show that (3x-2) is a factor of (3x^3+x^2-20x+12) through long or synthetic division. Here I will show you my attempt at a proof.\r
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\n" ); document.write( "\n" ); document.write( "I will prove this in the most elegant fashion I can.\r
\n" ); document.write( "\n" ); document.write( "Proof. We use a direct proof.
\n" ); document.write( "Predicate: (3x-2) is a factor of 3x^3+x^2-20x+12.\r
\n" ); document.write( "\n" ); document.write( "By definition, a cubic, if factorisable, will factor as (ax+b)(cx^2+dx+e), with {a, b, c, d, e} particular constants.\r
\n" ); document.write( "\n" ); document.write( "So for our predicate, in order for (3x-2) to be a factor, we need (3x-2)(cx^2+dx+e).\r
\n" ); document.write( "\n" ); document.write( "Moreover, (3x-2)(cx^2+dx+e)=3x^3+x^2-20x+12
\n" ); document.write( "and, -2e-2dx+3ex-2cx^2+3dx^2+3cx^3=3x^3+x^2-20x+12.\r
\n" ); document.write( "\n" ); document.write( "By definition, polynomials are equivalent if their variables' coefficients are equal.\r
\n" ); document.write( "\n" ); document.write( "Thus:
\n" ); document.write( "-2e=12 implies e=-6
\n" ); document.write( "-2d+3e=-20
\n" ); document.write( "-2c+3d=1
\n" ); document.write( "3c=3 implies c=1\r
\n" ); document.write( "\n" ); document.write( "And it follows that:
\n" ); document.write( "-2d-18=-20 implies d=1.\r
\n" ); document.write( "\n" ); document.write( "This gives a factorization of (3x-2)(x^2+x-6)=(3x-2)(x-2)(x+3). Thus we are certain that (3x-2) is a factor of polynomial (3x^3+x^2-20x+12). QED \n" ); document.write( "