document.write( "Question 714299: I've been trying to factor this all night but cannot figure it out. Please help...16z^2-54z+35 I've tried AC=560, factors 40 and 14 but that doesn't account for the -54, it's positive 54. \n" ); document.write( "

Algebra.Com's Answer #438750 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"16z%5E2-54z%2B35\"$ , we can see that the first coefficient is $\"16\"$ , the second coefficient is $\"-54\"$ , and the last term is $\"35\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"16\"$ by the last term $\"35\"$ to get $\"%2816%29%2835%29=560\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"560\"$ (the previous product) and add to the second coefficient $\"-54\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"560\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"560\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,7,8,10,14,16,20,28,35,40,56,70,80,112,140,280,560\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-7,-8,-10,-14,-16,-20,-28,-35,-40,-56,-70,-80,-112,-140,-280,-560\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. 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 $\"560\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*560 = 560
\n" ); document.write( "2*280 = 560
\n" ); document.write( "4*140 = 560
\n" ); document.write( "5*112 = 560
\n" ); document.write( "7*80 = 560
\n" ); document.write( "8*70 = 560
\n" ); document.write( "10*56 = 560
\n" ); document.write( "14*40 = 560
\n" ); document.write( "16*35 = 560
\n" ); document.write( "20*28 = 560
\n" ); document.write( "(-1)*(-560) = 560
\n" ); document.write( "(-2)*(-280) = 560
\n" ); document.write( "(-4)*(-140) = 560
\n" ); document.write( "(-5)*(-112) = 560
\n" ); document.write( "(-7)*(-80) = 560
\n" ); document.write( "(-8)*(-70) = 560
\n" ); document.write( "(-10)*(-56) = 560
\n" ); document.write( "(-14)*(-40) = 560
\n" ); document.write( "(-16)*(-35) = 560
\n" ); document.write( "(-20)*(-28) = 560\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-54\"$ :\r
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First Number	Second Number	Sum
1	560	1+560=561
2	280	2+280=282
4	140	4+140=144
5	112	5+112=117
7	80	7+80=87
8	70	8+70=78
10	56	10+56=66
14	40	14+40=54
16	35	16+35=51
20	28	20+28=48
-1	-560	-1+(-560)=-561
-2	-280	-2+(-280)=-282
-4	-140	-4+(-140)=-144
-5	-112	-5+(-112)=-117
-7	-80	-7+(-80)=-87
-8	-70	-8+(-70)=-78
-10	-56	-10+(-56)=-66
-14	-40	-14+(-40)=-54
-16	-35	-16+(-35)=-51
-20	-28	-20+(-28)=-48

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-14\"$ and $\"-40\"$ add to $\"-54\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-14\"$ and $\"-40\"$ both multiply to $\"560\"$ and add to $\"-54\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-54z\"$ with $\"-14z-40z\"$ . Remember, $\"-14\"$ and $\"-40\"$ add to $\"-54\"$ . So this shows us that $\"-14z-40z=-54z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"16z%5E2%2Bhighlight%28-14z-40z%29%2B35\"$ Replace the second term $\"-54z\"$ with $\"-14z-40z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2816z%5E2-14z%29%2B%28-40z%2B35%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2z%288z-7%29%2B%28-40z%2B35%29\"$ Factor out the GCF $\"2z\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2z%288z-7%29-5%288z-7%29\"$ Factor out $\"5\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%282z-5%29%288z-7%29\"$ Combine like terms. Or factor out the common term $\"8z-7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"16z%5E2-54z%2B35\"$ factors to $\"%282z-5%29%288z-7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"16z%5E2-54z%2B35=%282z-5%29%288z-7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282z-5%29%288z-7%29\"$ to get $\"16z%5E2-54z%2B35\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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