What is the remainder of the synthetic division problem below

what is the remainder of the synthetic division problem below

What is the remainder of the synthetic division problem below?

To determine the remainder of a synthetic division problem, we follow a specific process. Let’s illustrate this with an example, assuming we want to divide a polynomial P(x) by a binomial of the form x - c.

Steps for Synthetic Division

  1. Write down the coefficients of the polynomial: For example, if we have P(x) = 2x^3 - 3x^2 + 4x - 5, we would write down the coefficients as ( [2, -3, 4, -5] ).

  2. Determine the value of c in x - c: If we are dividing by x - 3, then c = 3.

  3. Set up the synthetic division table:

    • Write the value of c on the left.
    • Write the coefficients of P(x) on the top row.

Here’s how it looks:

  3 |  2  -3   4  -5
  1. Carry down the leading coefficient (the first number in the row of coefficients):
    • In this example, it’s 2.
  3 |  2  -3   4  -5
      ----------------
         2
  1. Multiply c by the value just written below the line and write the result under the next coefficient:
    • 3 \times 2 = 6.
  3 |  2  -3   4  -5
      ----------------
         2   6
  1. Add the values in this column and write the result beneath the line:
    • -3 + 6 = 3.
  3 |  2  -3   4  -5
      ----------------
         2   3
  1. Repeat the process:
    • Multiply c by the value just obtained (3) and write the result under the next coefficient.
    • 3 \times 3 = 9.
  3 |  2  -3   4  -5
      ----------------
         2   3   9
  • Add the values in this column: 4 + 9 = 13.
  3 |  2  -3   4  -5
      ----------------
         2   3   13
  • Multiply c by the value just obtained (13) and write the result under the next coefficient.
  • 3 \times 13 = 39.
  3 |  2  -3   4  -5
      ----------------
         2   3   13  39
  • Add the values in this column: -5 + 39 = 34.
  3 |  2  -3   4  -5
      ----------------
         2   3   13  34

The last value at the bottom is the remainder.

Answer:

In this specific example of dividing ( 2x^3 - 3x^2 + 4x - 5 ) by ( x - 3 ), the remainder is 34.

If you provide the specific polynomial and divisor for your problem, I can apply the same steps to find the remainder.