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Use the remainder theorem to find the value of P(x) = 3ˣ³ - 5ˣ² - 4x + 4 when the remainder is 2.

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Final answer:

The remainder theorem states that the remainder of dividing a polynomial P(x) by x - r is P(r). Without a specific value for r, we cannot find for which x, P(x) equals 2. However, the steps to use the theorem with a given divisor were provided.

Step-by-step explanation:

To use the remainder theorem to find the value of P(x) when the remainder is 2, we need to identify a value of x such that upon substituting into P(x), the remainder from the division is exactly 2. The remainder theorem states that if a polynomial P(x) is divided by x - r, the remainder is P(r). So if the remainder is 2, then P(r) = 2 for some number r.

Without additional information to specify the value of r or the divisor, we cannot determine the exact value of x for which P(x) = 2. Normally, you would find such an r by testing possible factors of the constant term, however, this question does not provide a specific divisor like x-c for application.

Still, the procedure to find x, given the divisor, is as follows:

  1. Divide the polynomial by x - c using synthetic division or long division.
  2. Find the value that makes the quotient multiplied by x - c plus the remainder (2 in this case) equal to P(x).
  3. Substitute this value into P(x) to confirm that the remainder is indeed 2.

User Thomas Uhrig
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