document.write( "Question 1090305: use the p over q method and synthetic division to factor the polynomial P(x). then solve P(x)=0\r
\n" ); document.write( "\n" ); document.write( "P(x)=x^3+4x^2+x-6 \n" ); document.write( "

Algebra.Com's Answer #704742 by MathLover1(20850) $\"\"$ $\"About$

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P(x)=x^3+4x^2+x-6 \r
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\n" ); document.write( "\n" ); document.write( "1. Arrange the polynomial in descending order: \r
\n" ); document.write( "\n" ); document.write( " $\"P%28x%29=x%5E3%2B4x%5E2%2Bx-6+\"$ \r
\n" ); document.write( "\n" ); document.write( "2. Write down all the factors of the constant term. These are all the possible values of p . \r
\n" ); document.write( "\n" ); document.write( " $\"p=6\"$ , all the factors are\r
\n" ); document.write( "\n" ); document.write( " $\"p\"$ = $\"1\"$ ,ą $\"2\"$ , $\"3\"$ , $\"6\"$ \r
\n" ); document.write( "\n" ); document.write( "3. Write down all the factors of the leading coefficient. These are all the possible values of q .\r
\n" ); document.write( "\n" ); document.write( " $\"q=1\"$ \r
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\n" ); document.write( "\n" ); document.write( "4.Write down all the possible values of $\"p%2Fq\"$ . Remember that since factors can be negative, $\"p%2Fq+\"$ and $\"-+p%2Fq\"$ must both be included. Simplify each value and cross out any duplicates. \r
\n" ); document.write( "\n" ); document.write( " $\"p%2Fq\"$ =ą $\"1%2F1\"$ , ą $\"2%2F1\"$ ,ą $\"3%2F1\"$ , ą $\"6%2F1\"$ ,... (eliminate $\"6%2F1\"$ , it is product of $\"2%2F1\"$ and $\"3%2F1\"$ )\r
\n" ); document.write( "\n" ); document.write( " $\"p%2Fq\"$ =ą $\"1\"$ , ą $\"2\"$ ,ą $\"3\"$ , \r
\n" ); document.write( "\n" ); document.write( "5. Use synthetic division to determine the values of $\"+p%2Fq\"$ for which $\"P%28p%2Fq%29+=+0\"$ . These are all the rational roots of $\"P%28x%29+\"$ . \r
\n" ); document.write( "\n" ); document.write( "The synthetic division table is: if $\"p%2Fq=+1\"$
\n" ); document.write( " $\"1\"$ |.... $\"1\"$ .... $\"+4\"$ .... .. $\"1\"$ .... $\"+-6\"$
\n" ); document.write( "................... $\"1\"$ .... .. $\"5\"$ .... .. $\"6\"$
\n" ); document.write( "-----------------------------------------
\n" ); document.write( "......... $\"1\"$ ........ $\"5\"$ ....... $\"6\"$ ........ $\"0\"$ -> reminder=> $\"0\"$ ,=> $\"highlight%281%29\"$ $\"is\"$ a root\r
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\n" ); document.write( "\n" ); document.write( "The synthetic division table is: if $\"p%2Fq=+-1\"$
\n" ); document.write( " $\"-1\"$ |.... $\"1\"$ .... $\"4\"$ .... .. $\"1\"$ .... $\"+-6\"$
\n" ); document.write( "................... $\"-1\"$ .... $\"-3\"$ .... .... $\"+2\"$
\n" ); document.write( "-----------------------------------------
\n" ); document.write( "......... $\"1\"$ ........ $\"3\"$ ...... $\"-2\"$ .... $\"-4\"$ -> reminder $\"-4\"$ => $\"-1\"$ is $\"not\"$ root\r
\n" ); document.write( "\n" ); document.write( "The synthetic division table is: if $\"p%2Fq=+2\"$
\n" ); document.write( " $\"2\"$ |.... $\"1\"$ .... $\"4\"$ .... .. $\"1\"$ .... $\"+-6\"$
\n" ); document.write( "................... $\"2\"$ .... .. $\"12\"$ .... $\"+26\"$
\n" ); document.write( "-----------------------------------------
\n" ); document.write( "......... $\"1\"$ ........ $\"6\"$ ....... $\"13\"$ ........ $\"20\"$ -> reminder=> $\"20\"$ ,=> $\"2\"$ is $\"not\"$ a root\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The synthetic division table is: if $\"p%2Fq=+-2\"$
\n" ); document.write( " $\"-2\"$ |.... $\"1\"$ .... $\"+4\"$ .... .. $\"1\"$ .... $\"+-6\"$
\n" ); document.write( "................... $\"-2\"$ .... .. $\"-4\"$ .... $\"6\"$
\n" ); document.write( "-----------------------------------------
\n" ); document.write( "......... $\"1\"$ ........ $\"2\"$ ....... $\"-3\"$ ....... $\"0\"$ -> reminder=> $\"0\"$ ,=> $\"highlight%28-2%29\"$ $\"is\"$ a root\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The synthetic division table is: if $\"p%2Fq=+3\"$
\n" ); document.write( " $\"3\"$ |.... $\"1\"$ .... $\"+4\"$ .... .. $\"1\"$ .... $\"+-6\"$
\n" ); document.write( "................... $\"3\"$ .... .. $\"21\"$ .... $\"66\"$
\n" ); document.write( "-----------------------------------------
\n" ); document.write( "......... $\"1\"$ ........ $\"7\"$ ....... $\"22\"$ ........ $\"60\"$ -> reminder=> $\"60\"$ ,=> $\"3\"$ is $\"not\"$ a root\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The synthetic division table is: if $\"p%2Fq=+-3\"$
\n" ); document.write( " $\"-3\"$ |.... $\"1\"$ .... $\"4\"$ .... .. $\"1\"$ .... $\"+-6\"$
\n" ); document.write( "................... $\"-3\"$ .... .. $\"-3\"$ .... $\"6\"$
\n" ); document.write( "-----------------------------------------
\n" ); document.write( "......... $\"1\"$ ........ $\"1\"$ ....... $\"-2\"$ ........ $\"0\"$ -> reminder=> $\"0\"$ ,=> $\"highlight%28-3%29\"$ $\"is\"$ a root\r
\n" ); document.write( "\n" ); document.write( " $\"P%28x%29=%28x+-+1%29+%28x+%2B+2%29+%28x+%2B+3%29\"$ \r
\n" ); document.write( "\n" ); document.write( "then solve $\"+P%28x%29=0+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"%28x+-+1%29+%28x+%2B+2%29+%28x+%2B+3%29=0\"$ \r
\n" ); document.write( "\n" ); document.write( "if $\"x+-+1=0\"$ => $\"x=1\"$
\n" ); document.write( "if $\"x+%2B+2=0\"$ => $\"x=-2\"$
\n" ); document.write( "if $\"x+%2B+3=0\"$ => $\"x=-3\"$ \r
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