document.write( "Question 211834: 2x^2+27x+56=0 please help factor \n" ); document.write( "

Algebra.Com's Answer #160046 by jim_thompson5910(35256) $\"\"$ $\"About$

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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"2x%5E2%2B27x%2B56\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"27\"$ , and the last term is $\"56\"$ .

Now multiply the first coefficient $\"2\"$ by the last term $\"56\"$ to get $\"%282%29%2856%29=112\"$ .

Now the question is: what two whole numbers multiply to $\"112\"$ (the previous product) and add to the second coefficient $\"27\"$ ?

To find these two numbers, we need to list all of the factors of $\"112\"$ (the previous product).

Factors of $\"112\"$ :

1,2,4,7,8,14,16,28,56,112

-1,-2,-4,-7,-8,-14,-16,-28,-56,-112

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to $\"112\"$ .

1*112 = 112
2*56 = 112
4*28 = 112
7*16 = 112
8*14 = 112
(-1)*(-112) = 112
(-2)*(-56) = 112
(-4)*(-28) = 112
(-7)*(-16) = 112
(-8)*(-14) = 112

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"27\"$ :

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First Number	Second Number	Sum
1	112	1+112=113
2	56	2+56=58
4	28	4+28=32
7	16	7+16=23
8	14	8+14=22
-1	-112	-1+(-112)=-113
-2	-56	-2+(-56)=-58
-4	-28	-4+(-28)=-32
-7	-16	-7+(-16)=-23
-8	-14	-8+(-14)=-22

From the table, we can see that there are no pairs of numbers which add to $\"27\"$ . So $\"2x%5E2%2B27x%2B56\"$ cannot be factored.

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Answer:

So $\"2%2Ax%5E2%2B27%2Ax%2B56\"$ doesn't factor at all (over the rational numbers).

So $\"2%2Ax%5E2%2B27%2Ax%2B56\"$ is prime.

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\n" ); document.write( "\n" ); document.write( "So make sure that you have the correct problem. \n" ); document.write( "

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