Respuesta :
Answer: Choice B. x-4
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Explanation:
We'll use the remainder theorem.
This theorem says that if you divide p(x) over (x-k), then the remainder is p(k).
We have p(x) = x^3 + 3x^2 - 16x - 48 and the denominator is one of the four choices A through D.
Let's go through each of them until we find the answer.
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For choice A, we have x+2 which is the same as x-(-2). It matches with x-k to show that k = -2.
Plug k = -2 into p(x) to get
p(x) = x^3 + 3x^2 - 16x - 48
p(-2) = (-2)^3 + 3(-2)^2 - 16(-2) - 48
p(-2) = -12
The remainder is -12, which is not 0 like we're after. We can cross A off the list.
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We'll follow the same ideas for choices B through D
For choice B, we plug in k = 4 because x-4 matches with x-k to show that k = 4.
p(x) = x^3 + 3x^2 - 16x - 48
p(4) = (4)^3 + 3(4)^2 - 16(4) - 48
p(4) = 0
We end up with a remainder of 0, so we can stop here. Dividing (x^3+3x^2-16x-48) all over (x-4) leads to some quotient and a remainder of 0.
Put another way: (x-4) is a factor of x^3+3x^2-16x-48
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Side notes:
- x^3+3x^2-16x-48 factors to (x-4)(x+4)(x+3)
- If you try out choice C, the remainder is -42
- The remainder for choice D is -240 (you'll use k = -8).
