document.write( "Question 715742: Use the rational root theorem to list all possible rational roots for each equation.then find any actual roots.\r
\n" ); document.write( "\n" ); document.write( "1). X^4-7x^2+12=0 \n" ); document.write( "

Algebra.Com's Answer #439605 by KMST(5328) $\"\"$ $\"About$

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Rational roots would be fractions (negative and positive)
\n" ); document.write( "whose numerator is a factor of the independent term, the constant $\"12\"$ ,
\n" ); document.write( "and whose denominator is a factor of the leading coefficient, the invisible $\"1\"$ in front of $\"x%5E4\"$ .
\n" ); document.write( "The denominator can only be $\"1\"$ , the only factor of $\"1\"$ ,
\n" ); document.write( "but $\"12\"$ has 6 factors:
\n" ); document.write( "1, 2, 3, 4, 6, and 12.
\n" ); document.write( "So the possible rational roots are:
\n" ); document.write( "-12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, and 12.
\n" ); document.write( "
\n" ); document.write( "Finding the roots is easier.
\n" ); document.write( "No need to try any of those rational roots.
\n" ); document.write( "If we change variables using $\"y=x%5E2\"$ ,
\n" ); document.write( "the equation becomes
\n" ); document.write( " $\"y%5E2-7y%2B12=0\"$ --> $\"%28y-3%29%28y-4%29=0\"$ and that factoring tells us that
\n" ); document.write( " $\"y=3\"$ and $\"y=4\"$ are solutions of $\"y%5E2-7y%2B12=0\"$
\n" ); document.write( "and will lead us to solutions of $\"x%5E4-7x%5E2%2B12=0\"$
\n" ); document.write( " $\"x%5E2=3\"$ leads us to $\"highlight%28x=-sqrt%283%29%29\"$ and $\"highlight%28x=sqrt%283%29%29\"$ (not rational roots< but at least real roots)
\n" ); document.write( " $\"x%5E2=4\"$ leads us to
\n" ); document.write( " $\"x=-sqrt%284%29\"$ --> $\"highlight%28x=-2%29\"$ and
\n" ); document.write( " $\"x=sqrt%284%29\"$ --> $\"highlight%28x=2%29\"$ (two of the 12 possible rational roots). \n" ); document.write( "

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