Answer :

Final answer:

By using synthetic division, it is determined that f(-2) equals 0 for the given function. Synthetic division steps include setting up coefficients, repeating the process of multiplication and addition until final value is obtained.

Explanation:

To evaluate f(-2) using synthetic division, you perform the following steps:

  • Set up the coefficients of your polynomial which are 2, -3, -23, 34, and 4 in a row.
  • Write -2 (which is what we're evaluating the function at) to the left and draw a synthetic division symbol, which looks like an upside-down long division symbol.

Now that setup is done, let's perform the synthetic division steps.

  1. Bring down the first coefficient (which is 2 in this case).
  2. Multiply this by -2 (which is the value we're evaluating at). Write the result below -3 and add those to get -7.
  3. Repeat this process, multiply -7 by -2 and write the result under -23, add them to get 9.
  4. Multiply 9 by -2, writing the result under 34, add them to get -2.
  5. Finally, multiply -2 by -2 and place the result under 4. Adding them gives 0.

The final row of numbers obtained from this synthetic division process represents the coefficients of the remainder polynomial and the final number, which is 0 in our case, is the value of the function at -2. Therefore, f(-2) = 0.

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