How do you find the remainder when dividing the polynomial f(x) = x^3 + 14x^2 + 51x + 22 by x – 7?

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To find the remainder when a polynomial is divided by a linear expression, we can use the Remainder Theorem. According to this theorem, the remainder of a polynomial f(x) when divided by x – k is simply f(k). In this case, we want to find the remainder of the polynomial f(x) = x^3 + 14x^2 + 51x + 22 when divided by x – 7.

Here are the steps to calculate the remainder:

  1. Identify k: Since we are dividing by x – 7, our value of k is 7.
  2. Substitute k into f(x): We will now calculate f(7).
  3. Calculate each term:
    • (7)^3 = 343
    • 14 * (7)^2 = 14 * 49 = 686
    • 51 * (7) = 357
    • 22 = 22
  4. Add the results: Now we combine all the calculated values:
    • f(7) = 343 + 686 + 357 + 22
  5. Compute the sum:
    • f(7) = 343 + 686 + 357 + 22 = 1408

Thus, the remainder when dividing the polynomial f(x) by x – 7 is 1408.

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