document.write( "Question 183030This question is from textbook College Algebra
\n" ); document.write( ": Help with Find all rational zeros of the polynomial (using synthetic division)
\n" ); document.write( "22. p(x)= $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "What I did: multiples of 1= +-1
\n" ); document.write( " multiples of 4= +-1 +-4 +-2
\n" ); document.write( "possible zeros: +-1 +-4 +-2 (after dividing constant over leading coeficient)
\n" ); document.write( "I used synthetic division to test which are zeros. +1 worked, +4 did not, +2 using the quotient of +1 (1 -1 -4 4) did not work gave me remainder of 8, but when I used the original coeficients (1 -2 -3 8 -4)it gave me a zero. I was told by my professor that either way it should work. \r
\n" ); document.write( "\n" ); document.write( "Any information is much appreciated, have at test coming up. Thank you \n" ); document.write( "

Algebra.Com's Answer #137401 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
Any rational zero can be found through this equation\r
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where p and q are the factors of the last and first coefficients\r
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\n" ); document.write( "\n" ); document.write( "So let's list the factors of -4 (the last coefficient):\r
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\r
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\n" ); document.write( "\n" ); document.write( "Now let's list the factors of 1 (the first coefficient):\r
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\r
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\n" ); document.write( "\n" ); document.write( "Now let's divide each factor of the last coefficient by each factor of the first coefficient\r
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\r
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\n" ); document.write( "\n" ); document.write( "Now simplify\r
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\n" ); document.write( "\n" ); document.write( "These are all the distinct rational zeros of the function that could occur (ie some of these values are NOT zeros, but could be)\r
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\r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"1\"$ is really a root for the function $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ given the possible zero $\"1\"$ :\r
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1	\|	1	-2	-3	8	-4
	\|		1	-1	-4	4
		1	-1	-4	4	0

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\n" ); document.write( "\n" ); document.write( "Since the remainder $\"0\"$ (the right most entry in the last row) is equal to zero, this means that $\"1\"$ is a zero of $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"2\"$ is really a root for the function $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ given the possible zero $\"2\"$ :\r
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2	\|	1	-2	-3	8	-4
	\|		2	0	-6	4
		1	0	-3	2	0

\r
\n" ); document.write( "\n" ); document.write( "Since the remainder $\"0\"$ (the right most entry in the last row) is equal to zero, this means that $\"2\"$ is a zero of $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"4\"$ is really a root for the function $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ given the possible zero $\"4\"$ :\r
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4	\|	1	-2	-3	8	-4
	\|		4	8	20	112
		1	2	5	28	108

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\n" ); document.write( "\n" ); document.write( "Since the remainder $\"108\"$ (the right most entry in the last row) is not equal to zero, this means that $\"4\"$ is not a zero of $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"-1\"$ is really a root for the function $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ given the possible zero $\"-1\"$ :\r
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-1	\|	1	-2	-3	8	-4
	\|		-1	3	0	-8
		1	-3	0	8	-12

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\n" ); document.write( "\n" ); document.write( "Since the remainder $\"-12\"$ (the right most entry in the last row) is not equal to zero, this means that $\"-1\"$ is not a zero of $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"-2\"$ is really a root for the function $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ given the possible zero $\"-2\"$ :\r
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-2	\|	1	-2	-3	8	-4
	\|		-2	8	-10	4
		1	-4	5	-2	0

\r
\n" ); document.write( "\n" ); document.write( "Since the remainder $\"0\"$ (the right most entry in the last row) is equal to zero, this means that $\"-2\"$ is a zero of $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"-4\"$ is really a root for the function $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ given the possible zero $\"-4\"$ :\r
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-4	\|	1	-2	-3	8	-4
	\|		-4	24	-84	304
		1	-6	21	-76	300

\r
\n" ); document.write( "\n" ); document.write( "Since the remainder $\"300\"$ (the right most entry in the last row) is not equal to zero, this means that $\"-4\"$ is not a zero of $\"x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So the rational zeros of $\"p%28x%29=x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ are: 1,2,-2\r
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\n" ); document.write( "\n" ); document.write( "In other words, if we plug in $\"x=1\"$ , $\"x=2\"$ or $\"x=-2\"$ into $\"p%28x%29=x%5E4-2x%5E3-3x%5E2%2B8x-4\"$ , we'll get 0 as a result (try it out if you aren't sure)\r
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\n" ); document.write( "\n" ); document.write( "Note: the zero 1 has a multiplicity of 2 (ie it is counted twice) \n" ); document.write( "

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