When polynomials are multiplied, all terms inside the brackets must be multiplied separately. Similarly, when dividing a polynomial, each term must be divided separately.

Example 1

Calculate the division [[$ \displaystyle\frac {16x^2 - 4x} {4x} $]].​

Method I

Divide each term of the polynomial [[$ 16x^2 - 4x $]] separately by the monomial [[$ 4x $]].

[[$ \displaystyle\frac {16x^2 - 4x} {4x} = \displaystyle\frac {16x^2} {4x} - \displaystyle\frac {4x} {4x} = 4x - 1 $]]​

Method II

The division calculation can also be accomplished by first converting the division into product form and then reducing it by the common terms of the numerator and denominator.

 

Note! If a polynomial that contains a multiplication is to be divided, only the multiplier and the multiplicand can be divided. If a division calculation were made for both factors, the division calculation should be performed twice.

If the polynomial is divided by a polynomial with at least two terms, the first method shown in Example 1 cannot be used. Simplifying such a fraction always requires dividing the numerator and denominator into product form parts that can be reduced.

The most common mistake when simplifying rational expressions containing polynomials is to remove individual terms from a polynomial that is presented in the sum form. In this case, no division calculation has been performed for each term to be divided. The sum should never be reduced!

Example 2

When a polynomial is divided by a polynomial, the expressions must first be displayed in product form.

 
Note! If -2 had been chosen as the common factor of the denominator, a common factor for the denominator and numerator could have been obtained immediately.