document.write( "Question 253899: Look at the trinomial below.\r
\n" ); document.write( "\n" ); document.write( "x^2 + 8x + 15\r
\n" ); document.write( "\n" ); document.write( "Factor the trinomial as the product of two binomials.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "I think the answer is (x+3)+ (x+5)
\n" ); document.write( "Is this correct.\r
\n" ); document.write( "\n" ); document.write( "Part B\r
\n" ); document.write( "\n" ); document.write( "Use what you know about binomials and trinomials to explain how you found
\n" ); document.write( "the answer.\r
\n" ); document.write( "\n" ); document.write( "I don't know how to answer this.
\n" ); document.write( "Part c\r
\n" ); document.write( "\n" ); document.write( "supppose the middle term of the trinomial had been -8x instead of 8x. How would
\n" ); document.write( "the factored form change? __________________________I think it would be
\n" ); document.write( "negative. \n" ); document.write( "

Algebra.Com's Answer #186250 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Looking at the expression $\"x%5E2%2B8x%2B15\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"8\"$ , and the last term is $\"15\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"15\"$ to get $\"%281%29%2815%29=15\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"15\"$ (the previous product) and add to the second coefficient $\"8\"$ ?\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 $\"15\"$ (the previous product).\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"15\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,5,15\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-15\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"15\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*15 = 15
\n" ); document.write( "3*5 = 15
\n" ); document.write( "(-1)*(-15) = 15
\n" ); document.write( "(-3)*(-5) = 15\r
\n" ); document.write( "
\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 $\"8\"$ :\r
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\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	15	1+15=16
3	5	3+5=8
-1	-15	-1+(-15)=-16
-3	-5	-3+(-5)=-8

\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"5\"$ add to $\"8\"$ (the middle coefficient).\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"5\"$ both multiply to $\"15\"$ and add to $\"8\"$ \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"8x\"$ with $\"3x%2B5x\"$ . Remember, $\"3\"$ and $\"5\"$ add to $\"8\"$ . So this shows us that $\"3x%2B5x=8x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%283x%2B5x%29%2B15\"$ Replace the second term $\"8x\"$ with $\"3x%2B5x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B3x%29%2B%285x%2B15%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3%29%2B%285x%2B15%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3%29%2B5%28x%2B3%29\"$ 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
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%2B5%29%28x%2B3%29\"$ Combine like terms. Or factor out the common term $\"x%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "===============================================================\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B8x%2B15\"$ factors to $\"%28x%2B5%29%28x%2B3%29\"$ .\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2B8x%2B15=%28x%2B5%29%28x%2B3%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B5%29%28x%2B3%29\"$ to get $\"x%5E2%2B8x%2B15\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "==========================================================================\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-8x%2B15\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-8\"$ , and the last term is $\"15\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"15\"$ to get $\"%281%29%2815%29=15\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"15\"$ (the previous product) and add to the second coefficient $\"-8\"$ ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"15\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"15\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,5,15\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-15\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"15\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*15 = 15
\n" ); document.write( "3*5 = 15
\n" ); document.write( "(-1)*(-15) = 15
\n" ); document.write( "(-3)*(-5) = 15\r
\n" ); document.write( "
\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 $\"-8\"$ :\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	15	1+15=16
3	5	3+5=8
-1	-15	-1+(-15)=-16
-3	-5	-3+(-5)=-8

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-3\"$ and $\"-5\"$ add to $\"-8\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-3\"$ and $\"-5\"$ both multiply to $\"15\"$ and add to $\"-8\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-8x\"$ with $\"-3x-5x\"$ . Remember, $\"-3\"$ and $\"-5\"$ add to $\"-8\"$ . So this shows us that $\"-3x-5x=-8x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-3x-5x%29%2B15\"$ Replace the second term $\"-8x\"$ with $\"-3x-5x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-3x%29%2B%28-5x%2B15%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x-3%29%2B%28-5x%2B15%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x-3%29-5%28x-3%29\"$ 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
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x-5%29%28x-3%29\"$ Combine like terms. Or factor out the common term $\"x-3\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "===============================================================\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-8x%2B15\"$ factors to $\"%28x-5%29%28x-3%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2-8x%2B15=%28x-5%29%28x-3%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x-5%29%28x-3%29\"$ to get $\"x%5E2-8x%2B15\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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