document.write( "Question 158061: Factor Completely\r
\n" ); document.write( "\n" ); document.write( "16x^4-40x^2+9 \n" ); document.write( "

Algebra.Com's Answer #116458 by jim_thompson5910(35256) $\"\"$ $\"About$

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Let $\"z=x%5E2\"$ . So this means that $\"z%5E2=x%5E4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So the expression goes from $\"16x%5E4-40x%5E2%2B9\"$ to $\"16z%5E2-40z%2B9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"16z%5E2-40z%2B9\"$ , we can see that the first coefficient is $\"16\"$ , the second coefficient is $\"-40\"$ , and the last term is $\"9\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"16\"$ by the last term $\"9\"$ to get $\"%2816%29%289%29=144\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"144\"$ (the previous product) and add to the second coefficient $\"-40\"$ ?\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 $\"144\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"144\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,16,18,24,36,48,72,144\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72,-144\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 $\"144\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*144
\n" ); document.write( "2*72
\n" ); document.write( "3*48
\n" ); document.write( "4*36
\n" ); document.write( "6*24
\n" ); document.write( "8*18
\n" ); document.write( "9*16
\n" ); document.write( "12*12
\n" ); document.write( "(-1)*(-144)
\n" ); document.write( "(-2)*(-72)
\n" ); document.write( "(-3)*(-48)
\n" ); document.write( "(-4)*(-36)
\n" ); document.write( "(-6)*(-24)
\n" ); document.write( "(-8)*(-18)
\n" ); document.write( "(-9)*(-16)
\n" ); document.write( "(-12)*(-12)\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 $\"-40\"$ :\r
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First Number	Second Number	Sum
1	144	1+144=145
2	72	2+72=74
3	48	3+48=51
4	36	4+36=40
6	24	6+24=30
8	18	8+18=26
9	16	9+16=25
12	12	12+12=24
-1	-144	-1+(-144)=-145
-2	-72	-2+(-72)=-74
-3	-48	-3+(-48)=-51
-4	-36	-4+(-36)=-40
-6	-24	-6+(-24)=-30
-8	-18	-8+(-18)=-26
-9	-16	-9+(-16)=-25
-12	-12	-12+(-12)=-24

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-4\"$ and $\"-36\"$ add to $\"-40\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"-36\"$ both multiply to $\"144\"$ and add to $\"-40\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-40z\"$ with $\"-4z-36z\"$ . Remember, $\"-4\"$ and $\"-36\"$ add to $\"-40\"$ . So this shows us that $\"-4z-36z=-40z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"16z%5E2%2Bhighlight%28-4z-36z%29%2B9\"$ Replace the second term $\"-40z\"$ with $\"-4z-36z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2816z%5E2-4z%29%2B%28-36z%2B9%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"4z%284z-1%29%2B%28-36z%2B9%29\"$ Factor out the GCF $\"4z\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"4z%284z-1%29-9%284z-1%29\"$ 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( " $\"%284z-9%29%284z-1%29\"$ Combine like terms. Or factor out the common term $\"4z-1\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%284x%5E2-9%29%284x%5E2-1%29\"$ Plug in $\"z=x%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%282x%2B3%29%282x-3%29%282x%2B1%29%282x-1%29\"$ Factor each group using the difference of squares. \r
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\n" ); document.write( "\n" ); document.write( "Note: $\"4x%5E2-9\"$ factors to $\"%282x%2B3%29%282x-3%29\"$ and $\"4x%5E2-1\"$ factors to $\"%282x%2B1%29%282x-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"16x%5E4-40x%5E2%2B9\"$ completely factors to $\"%282x%2B3%29%282x-3%29%282x%2B1%29%282x-1%29\"$ .\r
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