Question 126457: (x+1)(x+2)(x+3)(x+4) = 120
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! An analytical way of doing this problem is to look at the factors of 120.
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Divide it by 2 and you get 60*2
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Divide the 60 by 4 to get 4 and 15 as factors of 60 and you have that the factors of 120
have become 4*15*2.
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Then factor 15 into 5*3 and substitute that result for 15 to get that the factors of 120
are 4*3*5*2. If you multiply these factors together you do get 120, so our factors check out
to be OK.
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Now arrange the factors in ascending order: 2*3*4*5
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Notice significantly that each of these factors increases by 1. Now compare them to the
factors on the left side of the problem you were given ...
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(x+1)*(x+2)*(x+3)*(x+4)
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Each of those factors is 1 greater than the preceding factor ... and there are four of them
just as we found in the four factors 2*3*4*5
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What do we get if we set the first factor x + 1 equal to 2 ... we find x = 1
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What do we get if we set the second factor x + 2 equal to 3 ... we again find x = 1
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What do we get if we set the third factor x + 3 equal to 4 ... we again find x = 1
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And, finally, what do we get if we set the fourth factor x + 4 equal to 5 ... we again find x = 1
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What an interesting outcome. If in each of the four factors:
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(x+1)*(x+2)*(x+3)*(x+4)
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we set x = 1, the factors become 2*3*4*5 and this product is 120. So the solution
to this problem is x = 1
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Sure was a lot easier to do it this way than to try multiplying out the 4 original factors
and then setting that product equal to 120 and trying to solve that fourth order equation.
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Hope this is clear enough so that you can follow it through and see what was done to solve
the problem.
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