document.write( "Question 150023: ) Consider the polynomial f(x) = x3 + 6x2 – 25x + 18.\r
\n" ); document.write( "\n" ); document.write( "Find all of the zeros of the given polynomial. Be sure to show work.\r
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Algebra.Com's Answer #110112 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "First, let's list all of the possible rational zeros.\r
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\n" ); document.write( "\n" ); document.write( "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 18 (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\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 $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ given the possible zero $\"1\"$ :\r
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1	\|	1	6	-25	18
	\|		1	7	-18
		1	7	-18	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 $\"1\"$ is a zero of $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "------------------------------------------------------\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 $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ given the possible zero $\"2\"$ :\r
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2	\|	1	6	-25	18
	\|		2	16	-18
		1	8	-9	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 $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "------------------------------------------------------\r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"3\"$ is really a root for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ given the possible zero $\"3\"$ :\r
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3	\|	1	6	-25	18
	\|		3	27	6
		1	9	2	24

\r
\n" ); document.write( "\n" ); document.write( "Since the remainder $\"24\"$ (the right most entry in the last row) is not equal to zero, this means that $\"3\"$ is not a zero of $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "------------------------------------------------------\r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"6\"$ is really a root for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ given the possible zero $\"6\"$ :\r
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6	\|	1	6	-25	18
	\|		6	72	282
		1	12	47	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 $\"6\"$ is not a zero of $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "------------------------------------------------------\r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"9\"$ is really a root for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ given the possible zero $\"9\"$ :\r
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9	\|	1	6	-25	18
	\|		9	135	990
		1	15	110	1008

\r
\n" ); document.write( "\n" ); document.write( "Since the remainder $\"1008\"$ (the right most entry in the last row) is not equal to zero, this means that $\"9\"$ is not a zero of $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "------------------------------------------------------\r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"18\"$ is really a root for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ given the possible zero $\"18\"$ :\r
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18	\|	1	6	-25	18
	\|		18	432	7326
		1	24	407	7344

\r
\n" ); document.write( "\n" ); document.write( "Since the remainder $\"7344\"$ (the right most entry in the last row) is not equal to zero, this means that $\"18\"$ is not a zero of $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "------------------------------------------------------\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 $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ given the possible zero $\"-1\"$ :\r
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-1	\|	1	6	-25	18
	\|		-1	-5	30
		1	5	-30	48

\r
\n" ); document.write( "\n" ); document.write( "Since the remainder $\"48\"$ (the right most entry in the last row) is not equal to zero, this means that $\"-1\"$ is not a zero of $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "------------------------------------------------------\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 $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ given the possible zero $\"-2\"$ :\r
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-2	\|	1	6	-25	18
	\|		-2	-8	66
		1	4	-33	84

\r
\n" ); document.write( "\n" ); document.write( "Since the remainder $\"84\"$ (the right most entry in the last row) is not equal to zero, this means that $\"-2\"$ is not a zero of $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"-3\"$ is really a root for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ given the possible zero $\"-3\"$ :\r
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-3	\|	1	6	-25	18
	\|		-3	-9	102
		1	3	-34	120

\r
\n" ); document.write( "\n" ); document.write( "Since the remainder $\"120\"$ (the right most entry in the last row) is not equal to zero, this means that $\"-3\"$ is not a zero of $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "------------------------------------------------------\r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"-6\"$ is really a root for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ given the possible zero $\"-6\"$ :\r
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-6	\|	1	6	-25	18
	\|		-6	0	150
		1	0	-25	168

\r
\n" ); document.write( "\n" ); document.write( "Since the remainder $\"168\"$ (the right most entry in the last row) is not equal to zero, this means that $\"-6\"$ is not a zero of $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "------------------------------------------------------\r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"-9\"$ is really a root for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ given the possible zero $\"-9\"$ :\r
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-9	\|	1	6	-25	18
	\|		-9	27	-18
		1	-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 $\"-9\"$ is a zero of $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "------------------------------------------------------\r
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\n" ); document.write( "\n" ); document.write( "Let's see if the possible zero $\"-18\"$ is really a root for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So let's make the synthetic division table for the function $\"x%5E3%2B6x%5E2-25x%2B18\"$ given the possible zero $\"-18\"$ :\r
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-18	\|	1	6	-25	18
	\|		-18	216	-3438
		1	-12	191	-3420

\r
\n" ); document.write( "\n" ); document.write( "Since the remainder $\"-3420\"$ (the right most entry in the last row) is not equal to zero, this means that $\"-18\"$ is not a zero of $\"x%5E3%2B6x%5E2-25x%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "==================================================\r
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\n" ); document.write( "\n" ); document.write( "Summary:\r
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\n" ); document.write( "\n" ); document.write( "So the rational zeros of $\"x%5E3%2B6x%5E2-25x%2B18\"$ are \r
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\n" ); document.write( "\n" ); document.write( " $\"x=-9\"$ , $\"x=1\"$ , or $\"x=2\"$ \r
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