How to Factor a Cubic Polynomial Using Synthetic Division?

Synthetic division provides a streamlined way to find roots of a cubic polynomial and subsequently factor it. Here's how to use it:

1. Find a Potential Root Using the Rational Root Theorem:

The Rational Root Theorem states that any rational root of the polynomial ax³ + bx² + cx + d = 0 will be of the form p/q, where p is a factor of the constant term d, and q is a factor of the leading coefficient a.
List all possible rational roots (both positive and negative).

2. Set Up the Synthetic Division:

Write the potential root you're testing (let's call it r) to the left, outside a division-like symbol.
Write the coefficients of the cubic polynomial ( a, b, c, and d) horizontally to the right of the division symbol. Make sure to include 0 for any missing terms.

3. Perform the Synthetic Division:

Bring down the leading coefficient (a) to the bottom row.
Multiply the number you just brought down (a) by the potential root (r). Write the result under the next coefficient (b).
Add the numbers in that column (b and the result from step 2). Write the sum in the bottom row.
Repeat steps 2 and 3 for the remaining coefficients.

4. Interpret the Results:

If the remainder (the last number in the bottom row) is 0, then the potential root r is a root of the polynomial. This means (x - r) is a factor of the polynomial.
If the remainder is not 0, then r is not a root, and you should try a different potential root.

5. Write the Factored Form:

If you found a root r, the numbers in the bottom row (excluding the last number, which is the remainder) are the coefficients of the quadratic factor. Specifically, if the bottom row contains a', b', and c', then the quadratic factor is a'x² + b'x + c'.
The factored form of the cubic polynomial is then (x - r)(a'x² + b'x + c').

6. Factor the Quadratic (if possible):

The quadratic factor a'x² + b'x + c' can sometimes be factored further into two linear factors. You can use factoring techniques, the quadratic formula, or complete the square.

Example:

Factor the cubic polynomial x³ - 6x² + 11x - 6.

1 | 1  -6  11  -6
  |      1  -5   6
  |----------------
    1  -5   6   0

Interpretation: The remainder is 0, so x = 1 is a root. Therefore, (x - 1) is a factor. The quadratic factor is x² - 5x + 6.
Write Factored Form: (x - 1)(x² - 5x + 6)
Factor Quadratic: x² - 5x + 6 = (x - 2)(x - 3)

Therefore, the fully factored form is (x - 1)(x - 2)(x - 3).

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