document.write( "Question 181934: These need to be factored completely\r
\n" ); document.write( "\n" ); document.write( "30z^8 + 44z^5 +16z^2 Could it be 2z^2(3z^ + 2)(5z^3 +4)\r
\n" ); document.write( "\n" ); document.write( "24x² + 14xy +2y² \r
\n" ); document.write( "\n" ); document.write( "(m+n)(x+3) + (m+n)(5+5) Could it be (m+n+3)(x+y+5)\r
\n" ); document.write( "\n" ); document.write( "Solve using the principal of zero products\r
\n" ); document.write( "\n" ); document.write( "(x+ 1/7)(x-4/5) = 0\r
\n" ); document.write( "\n" ); document.write( "Find the x-intercepts for the graph of the equation\r
\n" ); document.write( "\n" ); document.write( "Y = x² + 4x -45 Could it be (-9,0,(5,0)\r
\n" ); document.write( "\n" ); document.write( "Factor by grouping
\n" ); document.write( "-36x² -30x + 36 Could it be -6(3x-2)(2x+3)
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Algebra.Com's Answer #136554 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'll do the first two, which will hopefully help you with the rest of the problems. If not, then repost.\r
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\n" ); document.write( "\n" ); document.write( " $\"30z%5E8%2B44z%5E5%2B16z%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2z%5E2%2815z%5E6%2B22z%5E3%2B8%29\"$ Factor out the GCF $\"2z%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"15z%5E6%2B22z%5E3%2B8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"15z%5E6%2B22z%5E3%2B8\"$ we can see that the first term is $\"15z%5E6\"$ and the last term is $\"8\"$ where the coefficients are 15 and 8 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 15 and the last coefficient 8 to get 120. Now what two numbers multiply to 120 and add to the middle coefficient 22? Let's list all of the factors of 120:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 120:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120 ...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 120\r
\n" ); document.write( "\n" ); document.write( "1*120\r
\n" ); document.write( "\n" ); document.write( "2*60\r
\n" ); document.write( "\n" ); document.write( "3*40\r
\n" ); document.write( "\n" ); document.write( "4*30\r
\n" ); document.write( "\n" ); document.write( "5*24\r
\n" ); document.write( "\n" ); document.write( "6*20\r
\n" ); document.write( "\n" ); document.write( "8*15\r
\n" ); document.write( "\n" ); document.write( "10*12\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-120)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-40)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(-12)\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 22? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 22\r
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First Number	Second Number	Sum
1	120	1+120=121
2	60	2+60=62
3	40	3+40=43
4	30	4+30=34
5	24	5+24=29
6	20	6+20=26
8	15	8+15=23
10	12	10+12=22
-1	-120	-1+(-120)=-121
-2	-60	-2+(-60)=-62
-3	-40	-3+(-40)=-43
-4	-30	-4+(-30)=-34
-5	-24	-5+(-24)=-29
-6	-20	-6+(-20)=-26
-8	-15	-8+(-15)=-23
-10	-12	-10+(-12)=-22

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 10 and 12 add up to 22 and multiply to 120\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"15z%5E6%2B22z%5E3%2B8\"$ , replace $\"22z%5E3\"$ with $\"10z%5E3%2B12z%5E3\"$ (notice $\"10z%5E3%2B12z%5E3\"$ adds up to $\"22z%5E3\"$ . So it is equivalent to $\"22z%5E3\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"15z%5E6%2Bhighlight%2810z%5E3%2B12z%5E3%29%2B8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"15z%5E6%2B10z%5E3%2B12z%5E3%2B8\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2815z%5E6%2B10z%5E3%29%2B%2812z%5E3%2B8%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"5z%5E3%283z%5E3%2B2%29%2B4%283z%5E3%2B2%29\"$ Factor out the GCF of $\"5z%5E3\"$ out of the first group. Factor out the GCF of $\"4\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%285z%5E3%2B4%29%283z%5E3%2B2%29\"$ Since we have a common term of $\"3z%5E3%2B2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"15z%5E6%2B10z%5E3%2B12z%5E3%2B8\"$ factors to $\"%285z%5E3%2B4%29%283z%5E3%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"15z%5E6%2B22z%5E3%2B8\"$ factors to $\"%285z%5E3%2B4%29%283z%5E3%2B2%29\"$ (since $\"15z%5E6%2B22z%5E3%2B8\"$ is equivalent to $\"15z%5E6%2B10z%5E3%2B12z%5E3%2B8\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"2z%5E2%2815z%5E6%2B22z%5E3%2B8%29\"$ and factors further to $\"2z%5E2%285z%5E3%2B4%29%283z%5E3%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"30z%5E8%2B44z%5E5%2B16z%5E2\"$ completely factors to $\"2z%5E2%285z%5E3%2B4%29%283z%5E3%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"24x%5E2%2B14xy%2B2y%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2%2812x%5E2%2B7xy%2By%5E2%29\"$ Factor out the GCF $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"12x%5E2%2B7xy%2By%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"12x%5E2%2B7xy%2By%5E2\"$ we can see that the first term is $\"12x%5E2\"$ and the last term is $\"y%5E2\"$ where the coefficients are 12 and 1 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 12 and the last coefficient 1 to get 12. Now what two numbers multiply to 12 and add to the middle coefficient 7? Let's list all of the factors of 12:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 12:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,12\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-12 ...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 12\r
\n" ); document.write( "\n" ); document.write( "1*12\r
\n" ); document.write( "\n" ); document.write( "2*6\r
\n" ); document.write( "\n" ); document.write( "3*4\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-6)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-4)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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First Number	Second Number	Sum
1	12	1+12=13
2	6	2+6=8
3	4	3+4=7
-1	-12	-1+(-12)=-13
-2	-6	-2+(-6)=-8
-3	-4	-3+(-4)=-7

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 3 and 4 add up to 7 and multiply to 12\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"12x%5E2%2B7xy%2By%5E2\"$ , replace $\"7xy\"$ with $\"3xy%2B4xy\"$ (notice $\"3xy%2B4xy\"$ adds up to $\"7xy\"$ . So it is equivalent to $\"7xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"12x%5E2%2Bhighlight%283xy%2B4xy%29%2By%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"12x%5E2%2B3xy%2B4xy%2By%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2812x%5E2%2B3xy%29%2B%284xy%2By%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%284x%2By%29%2By%284x%2By%29\"$ Factor out the GCF of $\"3x\"$ out of the first group. Factor out the GCF of $\"y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%2By%29%284x%2By%29\"$ Since we have a common term of $\"4x%2By\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"12x%5E2%2B3xy%2B4xy%2By%5E2\"$ factors to $\"%283x%2By%29%284x%2By%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"12x%5E2%2B7xy%2By%5E2\"$ factors to $\"%283x%2By%29%284x%2By%29\"$ (since $\"12x%5E2%2B7xy%2By%5E2\"$ is equivalent to $\"12x%5E2%2B3xy%2B4xy%2By%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"2%2812x%5E2%2B7xy%2By%5E2%29\"$ and factors further to $\"2%283x%2By%29%284x%2By%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"24x%5E2%2B14xy%2B2y%5E2\"$ completely factors to $\"2%283x%2By%29%284x%2By%29\"$ \r
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