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What does a remainder mean in synthetic division?

Published in Synthetic Division 3 mins read

In synthetic division, the remainder represents the value of the polynomial being divided evaluated at the value used for the division. Essentially, if you divide a polynomial f(x) by x - c using synthetic division, the remainder is f(c).

Explanation

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x - c. The process provides both the quotient and the remainder of the division. The remainder is crucial because it directly relates to the Remainder Theorem.

  • Remainder Theorem: This theorem states that if a polynomial f(x) is divided by x - c, then the remainder is f(c). This means substituting 'c' into the original polynomial will yield the remainder obtained through synthetic division.

How it works

  1. Setup: You set up the synthetic division using the value 'c' and the coefficients of the polynomial.
  2. Process: You perform the synthetic division steps, which involve bringing down the leading coefficient, multiplying by 'c', adding to the next coefficient, and repeating.
  3. Remainder: The last number you obtain in the bottom row is the remainder.

Example

Let's say we have the polynomial f(x) = x2 + 3x - 5, and we want to divide it by x - 2. We'll use synthetic division with c = 2.

2 |  1   3   -5
   |      2   10
   |----------------
     1   5    5

In this case, the remainder is 5. According to the Remainder Theorem, f(2) should also be 5. Let's check:

f(2) = (2)2 + 3(2) - 5 = 4 + 6 - 5 = 5

As expected, f(2) = 5, which is the same as the remainder we obtained through synthetic division. Therefore, the remainder tells us the functional value of the polynomial at the point x = c.

Significance

Understanding the meaning of the remainder in synthetic division is useful for:

  • Evaluating polynomials: It provides a quick way to find the value of a polynomial at a specific point.
  • Finding roots of polynomials: If the remainder is 0, then 'c' is a root of the polynomial, meaning f(c) = 0.
  • Factor Theorem: A direct consequence of the Remainder Theorem. It states that x - c is a factor of f(x) if and only if f(c) = 0. In other words, if the remainder is 0, then x-c is a factor.

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