document.write( "Question 127518This question is from textbook College Algebra
\n" ); document.write( ": Use the factor theorem to decide whether or not the second polynomial is a factor of the first.
\n" ); document.write( "9) 4x^2 - 33x + 65; x - 5 \n" ); document.write( "

Algebra.Com's Answer #93452 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"4x%5E2-33x%2B65\"$ we can see that the first term is $\"4x%5E2\"$ and the last term is $\"65\"$ where the coefficients are 4 and 65 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 4 and the last coefficient 65 to get 260. Now what two numbers multiply to 260 and add to the middle coefficient -33? Let's list all of the factors of 260:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 260:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,10,13,20,26,52,65,130,260\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-10,-13,-20,-26,-52,-65,-130,-260 ...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 260\r
\n" ); document.write( "\n" ); document.write( "1*260\r
\n" ); document.write( "\n" ); document.write( "2*130\r
\n" ); document.write( "\n" ); document.write( "4*65\r
\n" ); document.write( "\n" ); document.write( "5*52\r
\n" ); document.write( "\n" ); document.write( "10*26\r
\n" ); document.write( "\n" ); document.write( "13*20\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-260)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-130)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-65)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-52)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(-26)\r
\n" ); document.write( "\n" ); document.write( "(-13)*(-20)\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 -33? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -33\r
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First Number	Second Number	Sum
1	260	1+260=261
2	130	2+130=132
4	65	4+65=69
5	52	5+52=57
10	26	10+26=36
13	20	13+20=33
-1	-260	-1+(-260)=-261
-2	-130	-2+(-130)=-132
-4	-65	-4+(-65)=-69
-5	-52	-5+(-52)=-57
-10	-26	-10+(-26)=-36
-13	-20	-13+(-20)=-33

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -13 and -20 add up to -33 and multiply to 260\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"4x%5E2-33x%2B65\"$ , replace $\"-33x\"$ with $\"-13x%2B-20x\"$ (notice $\"-13x%2B-20x\"$ adds up to $\"-33x\"$ . So it is equivalent to $\"-33x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%5E2%2Bhighlight%28-13x%2B-20x%29%2B65\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"4x%5E2-13x-20x%2B65\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%284x%5E2-13x%29%2B%28-20x%2B65%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%284x-13%29-5%284x-13%29\"$ Factor out the GCF of $\"x\"$ 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( " $\"%28x-5%29%284x-13%29\"$ Since we have a common term of $\"4x-13\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2-13x-20x%2B65\"$ factors to $\"%28x-5%29%284x-13%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"4x%5E2-33x%2B65\"$ factors to $\"%28x-5%29%284x-13%29\"$ (since $\"4x%5E2-33x%2B65\"$ is equivalent to $\"4x%5E2-13x-20x%2B65\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2-33x%2B65\"$ factors to $\"%28x-5%29%284x-13%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "---------------------\r
\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x-5\"$ is a factor of $\"4x%5E2-33x%2B65\"$ \n" ); document.write( "

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