When multiplying integers, if there is an odd number of negative factors, then the product is?
2 answers:
Answer:
The product of an odd number of negative factors is negative.
Step-by-step explanation:
The factor -1 can be factored from any negative number. Then the product of some number of negative numbers is the product of that number of positive numbers and -1 raised to that power.
An even power of -1 is 1.
An odd power of -1 is -1. So, the product of an odd number of negative factors is negative.
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<u>Examples</u>
- (-2)(-3)(-4)(5) = (2)(3)(4)(5)(-1)³ = -120 . . . . the product of an odd number of negative factors is negative
- (-2)(-3)(4)(5) = (2)(3)(4)(5)(-1)² = 120 . . . . . the product of an even number of negative factors is positive
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<em>Comment on this result</em>
This is helpful to remember when considering the solutions to polynomial inequalities. Once the factored form of the polynomial is found, the solution will depend on the signs of the factors. For example, consider ...
(x +4)(x -2)(x -3) < 0
The zeros of these factors are -4, 2, 3. At each of these values of x, the sign of one of the factors will change. For x<-4, all three factors are negative, so the inequality is true. For -4<x<2, only two of the factors are negative, so the inequality is false. For 2<x<3, one of the factors is negative, so the inequality is again true. For 3<x, all factors are positive, so the inequality is false. Then the solution is (x < -4) ∪ (2 < x < 3).
See the graph for a plot of this polynomial and the solution to the inequality.
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