document.write( "Question 825381: Find all the real zeros of the function.\r
\n" ); document.write( "\n" ); document.write( "h(x)= $\"x%5E4-4x%5E3-9x%5E2%2B16x%2B20\"$ \n" ); document.write( "

Algebra.Com's Answer #497311 by KMST(5328) $\"\"$ $\"About$

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As per the rational zero theorem,
\n" ); document.write( "the possible rational zeros of $\"h%28x%29\"$ are integers factors of $\"20\"$ , meaning
\n" ); document.write( "-20, -5, -4, -2, -1, 1, 2, 4, 5, and 20.
\n" ); document.write( "
\n" ); document.write( "We can try them all, starting by the easiest ones.
\n" ); document.write( "Trying 1 and -1, we find $\"x=-1\"$ is one zero:
\n" ); document.write( " $\"h%281%29=1-4-9%2B16%2B20=-3-9%2B16%2B20=-12%2B16%2B20=4%2B20=24\"$
\n" ); document.write( " $\"h%28-1%29=1%2B4-9-16%2B20=5-9-16%2B20=-4-16%2B20=-20%2B20=0\"$
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\n" ); document.write( "Since $\"x=-1\"$ is a zero of $\"h%28x%29\"$ , $\"x-%28-1%29=x%2B1\"$ must be a factor of $\"h%28x%29\"$ .
\n" ); document.write( "That means that $\"h%28x%29\"$ divides evenly by $\"x%2B1\"$ , with no remainder.
\n" ); document.write( "Dividing $\"h%28x%29\"$ by $\"x%2B1\"$ we get
\n" ); document.write( " $\"Q%28x%29=x%5E3-5x%5E2-4x%2B20\"$ , so
\n" ); document.write( " $\"h%28x%29=%28x%2B1%29Q%28x%29\"$ .
\n" ); document.write( "(The division can be done by \"synthetic division\", or whatever method you have been taught in class).
\n" ); document.write( "
\n" ); document.write( "As per the rational zero theorem,
\n" ); document.write( "the possible rational zeros of $\"Q%28x%29\"$ are integers factors of $\"20\"$ , meaning
\n" ); document.write( "-20, -5, -4, -2, -1, 1, 2, 4, 5, and 20.
\n" ); document.write( "We know that $\"Q%281%29%3C%3E0\"$ because otherwise

\n" ); document.write( "We can find that $\"Q%282%29=0\"$ either by substituting 2 for x,
\n" ); document.write( "or by dividing $\"Q%28x%29\"$ by $\"x-2\"$ .
\n" ); document.write( " $\"Q%282%29=2%5E3-5%2A2%5E2-4%2A2%2B20=8-5%2A4-8%2B20=8-20-8%2B20=0\"$ ,
\n" ); document.write( "and dividing we find that
\n" ); document.write( " $\"Q%28x%29=%28x-2%29%28x%5E2-3x-10%29\"$ .
\n" ); document.write( "
\n" ); document.write( "So,
\n" ); document.write( " $\"h%28x%29=%28x%2B1%29Q%28x%29=%28x%2B1%29%28x-2%29%28x%5E2-3x-10%29\"$ .
\n" ); document.write( "
\n" ); document.write( "Quadratic polynomial $\"x%5E2-3x-10\"$ is easy to factor as
\n" ); document.write( " $\"x%5E2-3x-10=%28x%2B2%29%28x-5%29\"$ , so
\n" ); document.write( "

.
\n" ); document.write( " $\"highlight%28h%28x%29=%28x%2B1%29%28x-2%29%28x%2B2%29%28x-5%29%29\"$
\n" ); document.write( "and $\"h%28x%29=0\"$ for
\n" ); document.write( " $\"highlight%28x=-2%29\"$ , $\"highlight%28x=-1%29\"$ , $\"highlight%28x=2%29\"$ , and $\"highlight%28x=5%29\"$ . \n" ); document.write( "

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