How Can Synthetic Division Be Used to Find Factors of Polynomial Functions?

Synthetic division is a streamlined method for dividing a polynomial by a linear factor of the form x - c. The key lies in the Remainder Theorem: if you divide a polynomial f(x) by x - c, the remainder is equal to f(c). Therefore, if the remainder is zero, x - c is a factor of f(x).

The Process: Checking for Factors

Here's how synthetic division helps determine if a linear expression is a factor:

1. Set up the synthetic division: Write the value of c (from the divisor x - c) outside the division symbol. List the coefficients of the polynomial to be divided inside the division symbol. Make sure the polynomial is in descending order of powers and include '0' as a coefficient for any missing terms.
2. Perform synthetic division: Bring down the first coefficient. Multiply it by c and write the result under the next coefficient. Add those two numbers. Repeat this process until you reach the last coefficient.
3. Interpret the result: The last number in the bottom row is the remainder. If the remainder is 0, then x - c is a factor of the original polynomial. The other numbers in the bottom row are the coefficients of the quotient polynomial.

Example

Let's say we want to determine if x - 2 is a factor of f(x) = x³ - 4x² + x + 6.

1. Set up:

   2 | 1  -4   1   6

2. Perform Synthetic Division:

   2 | 1  -4   1   6
     |      2  -4  -6
     |----------------
       1  -2  -3   0

3. Interpret: Since the remainder is 0, x - 2 is a factor of x³ - 4x² + x + 6. The quotient is x² - 2x - 3. We can now write:

   x³ - 4x² + x + 6 = (x - 2)(x² - 2x - 3)

Furthermore, we can factor the quadratic quotient to get x³ - 4x² + x + 6 = (x - 2)(x - 3)(x + 1).

Benefits of Using Synthetic Division for Factoring

• Efficiency: Synthetic division is generally faster and easier than long division, especially for linear divisors.
• Remainder Theorem Application: It directly applies the Remainder Theorem to quickly identify factors.
• Quotient Identification: It provides the quotient polynomial, allowing you to further factor the polynomial.

When It's Not a Factor

If the remainder is not zero, then x - c is not a factor of f(x). The non-zero remainder represents the value of f(c).