document.write( "Question 117393: . If x = −2 is a zero of f (x) = x3 + 6x2 + 11x + 6, then f (x) factors
\n" ); document.write( "completely as ( use synthetic or long division; show your work) (LO 12)
\n" ); document.write( "a) (x − 2)(x + 3)(x + 1);
\n" ); document.write( "b) (x + 2)(x + 3)(x − 1);
\n" ); document.write( "c) (x − 2)(x + 3)(x − 1);
\n" ); document.write( "d) (x + 2)(x + 3)(x + 1);
\n" ); document.write( "I am lost with the synthetic division \n" ); document.write( "

Algebra.Com's Answer #85407 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Let's set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of $\"x%5E3+%2B+6x%5E2+%2B+11x+%2B+6\"$ to the right of the test zero.\n" ); document.write( "\n" ); document.write( "

-2	\|	1	6	11	6
	\|

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\n" ); document.write( "\n" ); document.write( "Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)\r
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-2	\|	1	6	11	6
	\|
		1

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\n" ); document.write( "\n" ); document.write( " Multiply -2 by 1 and place the product (which is -2) right underneath the second coefficient (which is 6)\r
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-2	\|	1	6	11	6
	\|		-2
		1

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\n" ); document.write( "\n" ); document.write( " Add -2 and 6 to get 4. Place the sum right underneath -2.\r
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-2	\|	1	6	11	6
	\|		-2
		1	4

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\n" ); document.write( "\n" ); document.write( " Multiply -2 by 4 and place the product (which is -8) right underneath the third coefficient (which is 11)\r
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-2	\|	1	6	11	6
	\|		-2	-8
		1	4

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\n" ); document.write( "\n" ); document.write( " Add -8 and 11 to get 3. Place the sum right underneath -8.\r
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-2	\|	1	6	11	6
	\|		-2	-8
		1	4	3

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\n" ); document.write( "\n" ); document.write( " Multiply -2 by 3 and place the product (which is -6) right underneath the fourth coefficient (which is 6)\r
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-2	\|	1	6	11	6
	\|		-2	-8	-6
		1	4	3

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\n" ); document.write( "\n" ); document.write( " Add -6 and 6 to get 0. Place the sum right underneath -6.\r
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-2	\|	1	6	11	6
	\|		-2	-8	-6
		1	4	3	0

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\n" ); document.write( "\n" ); document.write( "Since the last column adds to zero, we have a remainder of zero. This means $\"x%2B2\"$ is a factor of $\"x%5E3+%2B+6x%5E2+%2B+11x+%2B+6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now lets look at the bottom row of coefficients:\r
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\n" ); document.write( "\n" ); document.write( "The first 3 coefficients (1,4,3) form the quotient\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2+%2B+4x+%2B+3\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"%28x%5E3+%2B+6x%5E2+%2B+11x+%2B+6%29%2F%28x%2B2%29=x%5E2+%2B+4x+%2B+3\"$ \r
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\n" ); document.write( "\n" ); document.write( "You can use this online polynomial division calculator to check your work\r
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\n" ); document.write( "\n" ); document.write( "Basically $\"x%5E3+%2B+6x%5E2+%2B+11x+%2B+6\"$ factors to $\"%28x%2B2%29%28x%5E2+%2B+4x+%2B+3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now lets break $\"x%5E2+%2B+4x+%2B+3\"$ down further\r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"x%5E2%2B4x%2B3\"$ we can see that the first term is $\"x%5E2\"$ and the last term is $\"3\"$ where the coefficients are 1 and 3 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient 3 to get 3. Now what two numbers multiply to 3 and add to the middle coefficient 4? Let's list all of the factors of 3:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 3:\r
\n" ); document.write( "\n" ); document.write( "1,3\r
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\n" ); document.write( "\n" ); document.write( "-1,-3 ...List the negative factors as well. This will allow us to find all possible combinations\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to 3\r
\n" ); document.write( "\n" ); document.write( "1*3\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-3)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4\r
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First Number	Second Number	Sum
1	3	1+3=4
-1	-3	-1+(-3)=-4

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 1 and 3 add up to 4 and multiply to 3\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"x%5E2%2B4x%2B3\"$ , replace $\"4x\"$ with $\"1x%2B3x\"$ (notice $\"1x%2B3x\"$ adds up to $\"4x\"$ . So it is equivalent to $\"4x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%281x%2B3x%29%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E2%2B1x%2B3x%2B3\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B1x%29%2B%283x%2B3%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B1%29%2B3%28x%2B1%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"3\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B3%29%28x%2B1%29\"$ Since we have a common term of $\"x%2B1\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B1x%2B3x%2B3\"$ factors to $\"%28x%2B3%29%28x%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E2%2B4x%2B3\"$ factors to $\"%28x%2B3%29%28x%2B1%29\"$ (since $\"x%5E2%2B4x%2B3\"$ is equivalent to $\"x%5E2%2B1x%2B3x%2B3\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B4x%2B3\"$ factors to $\"%28x%2B3%29%28x%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B2%29%28x%2B3%29%28x%2B1%29\"$ Now reintroduce the first factor $\"x%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E3+%2B+6x%5E2+%2B+11x+%2B+6\"$ factors to $\"%28x%2B2%29%28x%2B3%29%28x%2B1%29\"$ which means the answer is D)\r
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\n" ); document.write( "\n" ); document.write( "Notice if we graph $\"x%5E3+%2B+6x%5E2+%2B+11x+%2B+6\"$ we get\r
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\n" ); document.write( "\n" ); document.write( " $\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E3+%2B+6x%5E2+%2B+11x+%2B+6%29+\"$ Graph of $\"x%5E3+%2B+6x%5E2+%2B+11x+%2B+6\"$ \r
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\n" ); document.write( "\n" ); document.write( "and if we graph $\"%28x%2B2%29%28x%2B3%29%28x%2B1%29\"$ , we get\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+%28x%2B2%29%28x%2B3%29%28x%2B1%29%29+\"$ Graph of $\"%28x%2B2%29%28x%2B3%29%28x%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "and you can see that the two graphs would overlap each other if they were plotted on the same screen. So this means that the two polynomials are equivalent. So this visually verifies our answer. \n" ); document.write( "

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