document.write( "Question 483522: Factor: 56b^2 + 23b -63 \n" ); document.write( "

Algebra.Com's Answer #330892 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"56b%5E2%2B23b-63\"$ , we can see that the first coefficient is $\"56\"$ , the second coefficient is $\"23\"$ , and the last term is $\"-63\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"56\"$ by the last term $\"-63\"$ to get $\"%2856%29%28-63%29=-3528\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-3528\"$ (the previous product) and add to the second coefficient $\"23\"$ ?\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 $\"-3528\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-3528\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,7,8,9,12,14,18,21,24,28,36,42,49,56,63,72,84,98,126,147,168,196,252,294,392,441,504,588,882,1176,1764,3528\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-7,-8,-9,-12,-14,-18,-21,-24,-28,-36,-42,-49,-56,-63,-72,-84,-98,-126,-147,-168,-196,-252,-294,-392,-441,-504,-588,-882,-1176,-1764,-3528\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 $\"-3528\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-3528) = -3528
\n" ); document.write( "2*(-1764) = -3528
\n" ); document.write( "3*(-1176) = -3528
\n" ); document.write( "4*(-882) = -3528
\n" ); document.write( "6*(-588) = -3528
\n" ); document.write( "7*(-504) = -3528
\n" ); document.write( "8*(-441) = -3528
\n" ); document.write( "9*(-392) = -3528
\n" ); document.write( "12*(-294) = -3528
\n" ); document.write( "14*(-252) = -3528
\n" ); document.write( "18*(-196) = -3528
\n" ); document.write( "21*(-168) = -3528
\n" ); document.write( "24*(-147) = -3528
\n" ); document.write( "28*(-126) = -3528
\n" ); document.write( "36*(-98) = -3528
\n" ); document.write( "42*(-84) = -3528
\n" ); document.write( "49*(-72) = -3528
\n" ); document.write( "56*(-63) = -3528
\n" ); document.write( "(-1)*(3528) = -3528
\n" ); document.write( "(-2)*(1764) = -3528
\n" ); document.write( "(-3)*(1176) = -3528
\n" ); document.write( "(-4)*(882) = -3528
\n" ); document.write( "(-6)*(588) = -3528
\n" ); document.write( "(-7)*(504) = -3528
\n" ); document.write( "(-8)*(441) = -3528
\n" ); document.write( "(-9)*(392) = -3528
\n" ); document.write( "(-12)*(294) = -3528
\n" ); document.write( "(-14)*(252) = -3528
\n" ); document.write( "(-18)*(196) = -3528
\n" ); document.write( "(-21)*(168) = -3528
\n" ); document.write( "(-24)*(147) = -3528
\n" ); document.write( "(-28)*(126) = -3528
\n" ); document.write( "(-36)*(98) = -3528
\n" ); document.write( "(-42)*(84) = -3528
\n" ); document.write( "(-49)*(72) = -3528
\n" ); document.write( "(-56)*(63) = -3528\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 $\"23\"$ :\r
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First Number	Second Number	Sum
1	-3528	1+(-3528)=-3527
2	-1764	2+(-1764)=-1762
3	-1176	3+(-1176)=-1173
4	-882	4+(-882)=-878
6	-588	6+(-588)=-582
7	-504	7+(-504)=-497
8	-441	8+(-441)=-433
9	-392	9+(-392)=-383
12	-294	12+(-294)=-282
14	-252	14+(-252)=-238
18	-196	18+(-196)=-178
21	-168	21+(-168)=-147
24	-147	24+(-147)=-123
28	-126	28+(-126)=-98
36	-98	36+(-98)=-62
42	-84	42+(-84)=-42
49	-72	49+(-72)=-23
56	-63	56+(-63)=-7
-1	3528	-1+3528=3527
-2	1764	-2+1764=1762
-3	1176	-3+1176=1173
-4	882	-4+882=878
-6	588	-6+588=582
-7	504	-7+504=497
-8	441	-8+441=433
-9	392	-9+392=383
-12	294	-12+294=282
-14	252	-14+252=238
-18	196	-18+196=178
-21	168	-21+168=147
-24	147	-24+147=123
-28	126	-28+126=98
-36	98	-36+98=62
-42	84	-42+84=42
-49	72	-49+72=23
-56	63	-56+63=7

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-49\"$ and $\"72\"$ add to $\"23\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-49\"$ and $\"72\"$ both multiply to $\"-3528\"$ and add to $\"23\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"23b\"$ with $\"-49b%2B72b\"$ . Remember, $\"-49\"$ and $\"72\"$ add to $\"23\"$ . So this shows us that $\"-49b%2B72b=23b\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"56b%5E2%2Bhighlight%28-49b%2B72b%29-63\"$ Replace the second term $\"23b\"$ with $\"-49b%2B72b\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2856b%5E2-49b%29%2B%2872b-63%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"7b%288b-7%29%2B%2872b-63%29\"$ Factor out the GCF $\"7b\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"7b%288b-7%29%2B9%288b-7%29\"$ Factor out $\"9\"$ 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( " $\"%287b%2B9%29%288b-7%29\"$ Combine like terms. Or factor out the common term $\"8b-7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"56b%5E2%2B23b-63\"$ factors to $\"%287b%2B9%29%288b-7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"56b%5E2%2B23b-63=%287b%2B9%29%288b-7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%287b%2B9%29%288b-7%29\"$ to get $\"56b%5E2%2B23b-63\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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