document.write( "Question 812405: Find a third degree polynomial function f(x) with real coefficients that has -3,-2,and 2 as zeros such that f(1)=-12 \n" ); document.write( "
Algebra.Com's Answer #489295 by jsmallt9(3758)\"\" \"About 
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A general formula for a 3rd degree polynomial in factored form would be:
\n" ); document.write( "\"f%28x%29=a%28x-z%5B1%5D%29%28x-z%5B2%5D%29%28x-z%5B3%5D%29\"
\n" ); document.write( "where the z's are zeros of the polynomial and the \"a\" is a non-zero constant. Using the given zeros we get:
\n" ); document.write( "\"f%28x%29=a%28x-%28-3%29%29%28x-%28-2%29%29%28x-2%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"f%28x%29=a%28x%2B3%29%28x%2B2%29%28x-2%29\"
\n" ); document.write( "Now we must multiply this out and find the right value for \"a\", either one first. First I will find the \"a\". For this we will use the fact that f(1) = -12:
\n" ); document.write( "\"-12=a%28%281%29%2B3%29%28%281%29%2B2%29%28%281%29-2%29+=+0\"
\n" ); document.write( "Simplifying...
\n" ); document.write( "\"-12=a%284%29%283%29%28-1%29\"
\n" ); document.write( "\"-12=-12a\"
\n" ); document.write( "Dividing by -12:
\n" ); document.write( "\"1+=+a\"
\n" ); document.write( "So:
\n" ); document.write( "\"f%28x%29=1%28x%2B3%29%28x%2B2%29%28x-2%29\"
\n" ); document.write( "Now we multiply this out. I'll start by using the \"%28a%2Bb%29%28a-b%29=a%5E2-b%5E2\" pattern to multiply the last two factors:
\n" ); document.write( "\"f%28x%29=%28x%2B3%29%28%28x%29%5E2-%282%29%5E2%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"f%28x%29=%28x%2B3%29%28x%5E2-4%29\"
\n" ); document.write( "Using FOIL on what's left:
\n" ); document.write( "\"f%28x%29=x%5E3-4x%2B3x%5E2-12\"
\n" ); document.write( "Reordering into standard form:
\n" ); document.write( "\"f%28x%29=x%5E3%2B3x%5E2-4x-12\" \n" ); document.write( "
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