What Is The Remainder For The Synthetic Division Problem Below

Understanding the Remainder in Synthetic Division: A Complete Guide

Synthetic division is a streamlined, powerful method for dividing polynomials, specifically when the divisor is a linear factor of the form (x - c). Its primary utility lies in its efficiency compared to traditional long division, but its most celebrated feature is the direct connection to the Remainder Theorem. This theorem provides an immediate answer to a fundamental question: what is the remainder for the synthetic division problem? The remainder is not just a byproduct of the calculation; it is the polynomial's value at x = c, offering profound insights into the polynomial's behavior and factors. Mastering this process unlocks a faster path to solving polynomial equations, analyzing functions, and understanding algebraic structures.

What is Synthetic Division?

Synthetic division is a shorthand algorithm for performing polynomial division. It is applicable only when dividing a polynomial P(x) by a linear binomial of the form (x - c), where c is a constant. The method eliminates the variables and cumbersome notation of long division, working solely with the coefficients of the polynomial in descending order of degree.

The setup is simple:

1. Write the coefficients of the dividend polynomial in a row.
2. Write the value c (from the divisor x - c) to the left.
3. Perform a series of multiplications and additions down the column. The final number in the bottom row is the remainder. All other numbers in that row represent the coefficients of the quotient polynomial, which will be one degree less than the original dividend.

Step-by-Step Process: Finding the Remainder

Let's walk through a concrete example. Suppose we want to find the remainder when P(x) = 2x³ - 6x² + 2x - 1 is divided by x - 3. Here, c = 3.

Step 1: Identify Coefficients and c. List the coefficients of P(x) in order: 2 (for x³), -6 (for x²), 2 (for x), -1 (constant). Note that we must include a zero for any missing degree. Our polynomial has all terms, so no zeros are needed. The value c = 3.

Step 2: Set Up the Synthetic Division Array.

3 | 2   -6    2    -1
  |_________________

Step 3: Perform the Algorithm.

• Bring down the first coefficient (2) directly below the line.
• Multiply this brought-down number (2) by c (3): 2 * 3 = 6. Write this product under the next coefficient (-6).
• Add the numbers in that column: -6 + 6 = 0. Write the sum (0) below the line.
• Repeat: Multiply the new sum (0) by c (3): 0 * 3 = 0. Write under the next coefficient (2).
• Add: 2 + 0 = 2. Write 2 below the line.
• Repeat: Multiply the new sum (2) by c (3): 2 * 3 = 6. Write under the last coefficient (-1).
• Add: -1 + 6 = 5. Write 5 below the line.

The completed array looks like this:

3 | 2   -6    2    -1
  |      6     0     6
  |___________________
  |     2     0     2   | 5

Step 4: Interpret the Result. The final number, 5, is the remainder. The other numbers (2, 0, 2) are the coefficients of the quotient polynomial: 2x² + 0x + 2, or simply 2x² + 2.

Therefore, the remainder for this synthetic division problem is 5.

The Scientific Foundation: The Remainder Theorem

The reason this process works is encapsulated in the Remainder Theorem. It states:

When a polynomial P(x) is divided by (x - c), the remainder is P(c).

This is not a coincidence; it is a direct algebraic consequence of the division algorithm: P(x) = (x - c) * Q(x) + R, where Q(x) is the quotient and R is the remainder. Since (x - c) is linear, R must be a constant. If we substitute x = c into this equation, the term (x - c) * Q(x) becomes zero, leaving P(c) = R.

Verifying our example: Calculate P(3) for P(x) = 2x³ - 6x² + 2x - 1. P(3) = 2*(3)³ - 6*(3)² + 2*(3) - 1