Discussion
When a polynomial \(P(x) = x^3 + 2x^2 - 5x + 7\) is divided by \((x-2)\), what is the remainder?
Answer: C
According to the Polynomial Remainder Theorem, if a polynomial P(x) is divided by \((x-a)\), the remainder is P(a). Here, the divisor is \((x-2)\), so a=2. We need to calculate P(2). \(P(2) = (2)^3 + 2(2)^2 - 5(2) + 7 = 8 + 2(4) - 10 + 7 = 8 + 8 - 10 + 7 = 16 - 10 + 7 = 6 + 7 = 13\). The remainder is 13.
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