Michael J. answered • 04/28/15
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We can set the product of the factors equal to zero. This will allow us to solve for x. The x values will be our reference values.
Set each factor equal to zero.
15 + 2x = 0 17 - 3x = 0 2 - 5x = 0 31 - 3x = 0
2x = -15 -3x = -17 -5x = -2 -3x = -31
x = -7.5 x = 5.67 x = 0.4 x = 10.33
The x values that will make the product of the factors equal to zero are
-7.5 , 0.4 , 5.67 , and 10.33
Next, we use test points around these x values to evaluate the product of the factors. We will use the following test points:
x = -8 , x = -2 , x = 2 , x = 7 , x = 11
Only some of these x values when plugged into the factors will give us a positive product. I will use negative and positive signs rather than numbers in my final calculations for each test point to show that the product is either positive or negative.
f(-8) = (15 + 2(-8)) * (17 - 3(-8)) * (2 - 5(-8)) * (31 - 3(-8))
= (15 - 16) * (17 + 24) * (2 + 40) * (31 + 24)
= - + + +
f(-2) = (15 + 2(-2)) * (17 - 3(-2)) * (2 - 5(-2)) * (31 - 3(-2))
= (15 - 4) * (17 + 6) * (2 + 10) * (31 + 6)
= + + + +
f(2) = (15 + 2(2)) * (17 - 3(2)) * (2 - 5(2)) * (31 - 3(2))
= (15 + 4) * (17 - 6) * (2 - 10) * (31 - 6)
= + + - +
f(7) = (15 + 2(7)) * (17 - 3(7)) * (2 - 5(7)) * (31 - 3(7))
= (15 + 14) * (17 - 21) * (2 - 35) * (31 - 21)
= + - - +
f(11) = (15 + 2(11)) * (17 - 3(11)) * (2 - 5(11)) * (31 - 3(11))
= (15 + 22) * (17 - 33) * (2 - 55) * (31 - 33)
= + - - -
The product is positive between x=-7.5 and x=0.4
The product is positive between x=5.67 and x=10.33
The integers of x where the product is positive are -7, -6, -5, -4, -2, -3, -2, -1, 0, 6, 7, 8, 9, 10
The answer to this question is 14 integers of x.
