How do you divide fractional polynomials?

Dividing fractional polynomials, especially when the denominator is a monomial, involves breaking the fraction into separate terms and simplifying each.

Dividing a Polynomial by a Monomial

The most straightforward case of dividing fractional polynomials occurs when you have a polynomial in the numerator and a monomial in the denominator. According to the provided reference, here's how you do it:

1. Break Up the Fraction: Divide each term in the numerator by the monomial in the denominator.

2. Simplify Each Term: Reduce each resulting fraction.

3. Check Your Work: Multiply the simplified result (quotient) by the original denominator (divisor). You should obtain the original numerator (dividend).

Example:

Let's say you want to divide (6x³ + 9x²) / (3x).

1. Break up the fraction:

   (6x³ / 3x) + (9x² / 3x)

2. Simplify each term:

   2x² + 3x

3. Check Your Work:

   (2x² + 3x) * (3x) = 6x³ + 9x² (This matches the original numerator.)

More Complex Cases

When the denominator is not a monomial but a more complex polynomial, you generally use polynomial long division or synthetic division (if the divisor is of the form x - a). These methods are similar to long division with numbers.

• Polynomial Long Division: Similar to numerical long division. Arrange both polynomials in descending order of exponents, and divide step by step.
• Synthetic Division: A shortcut method for dividing by a linear factor (x - a).

While polynomial long division and synthetic division are important, the initial question focuses on simpler fractional polynomial division (polynomial divided by a monomial) as directly addressed in the provided reference.