document.write( "Question 156977: How do I factor
\n" ); document.write( "16z to the 5th power + 12z to the fourth power - 10z to the third power \n" ); document.write( "

Algebra.Com's Answer #115742 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"16z%5E5%2B12z%5E4-10z%5E3\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2z%5E3%288z%5E2%2B6z-5%29\"$ Factor out the GCF $\"2z%5E3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"8z%5E2%2B6z-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"8z%5E2%2B6z-5\"$ we can see that the first term is $\"8z%5E2\"$ and the last term is $\"-5\"$ where the coefficients are 8 and -5 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 8 and the last coefficient -5 to get -40. Now what two numbers multiply to -40 and add to the middle coefficient 6? Let's list all of the factors of -40:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -40:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,20,40\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-20,-40 ...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 -40\r
\n" ); document.write( "\n" ); document.write( "(1)*(-40)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-10)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-8)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(40)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(20)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(10)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(8)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 6\r
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First Number	Second Number	Sum
1	-40	1+(-40)=-39
2	-20	2+(-20)=-18
4	-10	4+(-10)=-6
5	-8	5+(-8)=-3
-1	40	-1+40=39
-2	20	-2+20=18
-4	10	-4+10=6
-5	8	-5+8=3

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -4 and 10 add up to 6 and multiply to -40\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"8z%5E2%2B6z-5\"$ , replace $\"6z\"$ with $\"-4z%2B10z\"$ (notice $\"-4z%2B10z\"$ adds up to $\"6z\"$ . So it is equivalent to $\"6z\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"8z%5E2%2Bhighlight%28-4z%2B10z%29%2B-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"8z%5E2-4z%2B10z-5\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%288z%5E2-4z%29%2B%2810z-5%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"4z%282z-1%29%2B5%282z-1%29\"$ Factor out the GCF of $\"4z\"$ out of the first group. Factor out the GCF of $\"5\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%284z%2B5%29%282z-1%29\"$ Since we have a common term of $\"2z-1\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"8z%5E2-4z%2B10z-5\"$ factors to $\"%284z%2B5%29%282z-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"8z%5E2%2B6z-5\"$ factors to $\"%284z%2B5%29%282z-1%29\"$ (since $\"8z%5E2%2B6z-5\"$ is equivalent to $\"8z%5E2-4z%2B10z-5\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"2z%5E3%288z%5E2%2B6z-5%29\"$ and factors further to $\"2z%5E3%284z%2B5%29%282z-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"16z%5E5%2B12z%5E4-10z%5E3\"$ completely factors to $\"2z%5E3%284z%2B5%29%282z-1%29\"$ \r
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