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P(x) is divisible by (x+1) and (x-1) , find P(x) if P(2) = 9.
I determined the answer by trial and error and P(x) turned out to be x^3 + x^2 - x - 1 , but is there any systematic method to apply here if the polynomial was more complex ? How can I really solve this algebraicly ?
TIA
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(x+1) | P(x) and (x-1) | P(x).
So:
P(x) = (x+1)(x-1)Q(x) = (x^2-1)Q(x).
P(2) = (3)Q(2) = 9, so Q(2) = 3.
Q(x) = x+1 statifies this.
P(x) = (x+1)(x-1)(x+1) = (x^2-1)(x+1) = x^3 + x^2 - x - 1
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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Note that there are multiple solutions:
Q(x) = 3, then:
P(x) = (x^2-1)(3) = 3x^2 - 3
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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Thanks a lot Ricky!
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