document.write( "Question 936620: For the following polynomials , find all real and complex roots .(hint: Use the rational root theorem to identify potential roots.
\n" ); document.write( "1- y(x) = x^3-1
\n" ); document.write( "2- h(x) = x^4 -1
\n" ); document.write( "3- g(x) = x^4 +2x^3-16x^2-2x+15
\n" ); document.write( "4- f(x) = x^4+ 2x^3 -16x^2 -2x +15
\n" ); document.write( "(note:what they mean by potential roots?) .Thanks
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Algebra.Com's Answer #570087 by MathLover1(20849) $\"\"$ $\"About$

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to identify POTENTIAL roots use THE RATIONAL ZERO THEOREM\r
\n" ); document.write( "\n" ); document.write( "If $\"f%28x%29\"$ = $\"a%5Bn%5Dx%5En\"$ + .... .+ $\"a%5B1%5Dx\"$ + $\"a%5B0%5D\"$ has integer coefficients, then every rational zero of $\"f%28x%29\"$ has the following form:\r
\n" ); document.write( "\n" ); document.write( " $\"p%2Fq=%28factors_of_constant_term_a%5B0%5D%29%2F%28factors_of_leading_coefficient_a%29\"$ \r
\n" ); document.write( "\n" ); document.write( "using that, we have:
\n" ); document.write( "1- $\"y%28x%29+=+x%5E3-1\"$
\n" ); document.write( "--> factor $\"1\"$ and $\"1\"$ :
\n" ); document.write( " $\"x+\"$ = ą $\"1%2F1\"$ which is $\"x\"$ = ą $\"1%2F1\"$ \r
\n" ); document.write( "\n" ); document.write( "2- $\"h%28x%29+=+x%5E4+-1\"$
\n" ); document.write( "--> factor $\"1\"$ and $\"1\"$ :
\n" ); document.write( " $\"x+\"$ = ą $\"1%2F1\"$ which is $\"x\"$ = ą $\"1%2F1\"$ \r
\n" ); document.write( "\n" ); document.write( "3- $\"g%28x%29+=+x%5E4+%2B2x%5E3-16x%5E2-2x%2B15\"$
\n" ); document.write( "--> factor $\"1\"$ , and $\"15\"$ :
\n" ); document.write( " $\"1\"$ X $\"1\"$ , $\"3\"$ , $\"5\"$ , $\"15\"$
\n" ); document.write( " $\"x\"$ = ą $\"1%2F1\"$ , ą $\"3%2F1\"$ ,ą $\"5%2F1\"$ ,ą $\"15%2F1\"$ , which is => $\"x\"$ = ą $\"1\"$ ,ą $\"3\"$ ,ą $\"5\"$ ,ą $\"15\"$
\n" ); document.write( "Test these zeros using synthetic division.
\n" ); document.write( "test $\"x=1\"$
\n" ); document.write( " $\"g%28x%29+=+x%5E4+%2B2x%5E3-16x%5E2-2x%2B15\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"1\"$ |--1-----2----..-16------...-2----15
\n" ); document.write( "---| ------1----..-1----..-1------17---..-15
\n" ); document.write( "---| 1---- 1----.. -17--------15------- 0\r
\n" ); document.write( "\n" ); document.write( "so, $\"x=1\"$ is the root\r
\n" ); document.write( "\n" ); document.write( "same way you check all other potential roots\r
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\n" ); document.write( "\n" ); document.write( "now find roots:\r
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\n" ); document.write( "\n" ); document.write( "1- $\"y%28x%29+=+x%5E3-1\"$ ........factor: set $\"y%28x%29=0\"$ and use $\"a%5E3+-+b%5E3+=+%28a+-b%29%28a%5E2+%2B+ab+%2B+b%5E2%29++\"$ ..(difference of $\"2\"$ cubes rule)\r
\n" ); document.write( "\n" ); document.write( " $\"0=+%28x-1%29%28x%5E2%2Bx%2B1%29\"$ \r
\n" ); document.write( "\n" ); document.write( "we already know one root: if $\"0=+%28x-1%29\"$ => $\"x=1\"$ \r
\n" ); document.write( "\n" ); document.write( "use quadratic formula to find other two roots from $\"0=+%28x%5E2%2Bx%2B1%29\"$ :\r
\n" ); document.write( "\n" ); document.write( " $\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+%28-1+%2B-+sqrt%28+1%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+%28-1+%2B-+sqrt%28+1-4+%29%29%2F2+\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+%28-1+%2B-+sqrt%28+-3+%29%29%2F2+\"$ \r
\n" ); document.write( "\n" ); document.write( "solution will be complex roots:\r
\n" ); document.write( "\n" ); document.write( " $\"x+=+-1%2F2+%2B+sqrt%28+-3+%29%2F2+\"$
\n" ); document.write( "and
\n" ); document.write( " $\"x+=+-1%2F2+-+sqrt%28+-3+%29%2F2+\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2Cx%5E3-1%29+\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "2- $\"h%28x%29+=+x%5E4+-1\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0+=+x%5E4+-1%5E4\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0+=+%28x%5E2%29%5E2+-%281%5E2%29%5E2\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0+=+%28x%5E2+-1%5E2%29%28x%5E2+%2B1%5E2%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"%28x-1%29%28x%2B1%29%28x%5E2%2B1%29=0+\"$ \r
\n" ); document.write( "\n" ); document.write( "roots:\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x-1%29=0+\"$ => $\"x=1\"$
\n" ); document.write( " $\"%28x%2B1%29=0+\"$ => $\"x=-1\"$ ......real roots\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B1%29=0+\"$ => $\"x%5E2=-1+\"$ => $\"x=sqrt%28-1%29+\"$ => $\"x=i+\"$ or $\"x=-i+\"$
\n" ); document.write( " $\"x=1\"$ .........complex roots\r
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C%28x-1%29%28x%2B1%29%28x%5E2%2B1%29%29+\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "3- $\"g%28x%29+=+x%5E4+%2B2x%5E3-16x%5E2-2x%2B15\"$ ...write $\"2x%5E3\"$ as $\"5x%5E3-3x%5E3\"$ and $\"-16x%5E2\"$ as $\"-15x%5E2-x%5E2\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0+=+x%5E4-x%5E2%2B5x%5E3-5x-3x%5E3-3x-15x%5E2-15\"$ ....group\r
\n" ); document.write( "\n" ); document.write( " $\"0+=+%28x%5E4-x%5E2%29%2B%285x%5E3-5x%29-%283x%5E3-3x%29-%2815x%5E2-15%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0+=+x%5E2%28x%5E2-1%29%2B5x%28x%5E2-1%29-3x%28x%5E2-1%29-15%28x%5E2-1%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0+=+%28x%5E2-1%29%28x%5E2%2B5x-3x-15%29\"$ ...group\r
\n" ); document.write( "\n" ); document.write( " $\"0+=+%28x-1%29%28x%2B1%29%28%28x%5E2-3x%29%2B%285x-15%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0+=+%28x-1%29%28x%2B1%29%28x%28x-3%29%2B5%28x-3%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0+=+%28x-1%29%28x%2B1%29%28x-3%29%28x%2B5%29\"$ \r
\n" ); document.write( "\n" ); document.write( "roots:
\n" ); document.write( " $\"0+=+%28x-1%29\"$ => $\"x=1\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0+=%28x%2B1%29\"$ => $\"x=-1\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0+=+%28x-3%29\"$ => $\"x=3\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0+=+%28x%2B5%29\"$ => $\"x=-5\"$ ........all roots are real roots\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+600%2C+600%2C+-25%2C+25%2C+-45%2C+25%2C+x%5E4%2B+2x%5E3+-16x%5E2+-2x+%2B15%29+\"$ \r
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\n" ); document.write( "\n" ); document.write( "4- $\"f%28x%29+=+x%5E4%2B+2x%5E3+-16x%5E2+-2x+%2B15+\"$ this is same as 3- \r
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