Dividing Polynomials

Example 1: Simplify each of the following division expressions and find the pattern involving the exponents.

These examples suggest the following rule.

Example 2: Simplify each of the following.

These examples suggest the following rule:

Generally, when simplifying expressions, write the final result without the use of negative exponents.

Example 3: Simplify each of the following.

To divide a monomial by another monomial, follow the procedure.

1. Divide the numerical coefficients.

2. Divide the variables.

3. Multiply the results together.

Example 4: Simplify the following expressions by dividing correctly.

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Example 5: Simplify the given expression by dividing correctly.

To divide a polynomial by a polynomial, a procedure similar to long division in arithmetic is used. The procedure calls for four steps: divide, multiply, subtract, bring down. This procedure is repeated until there is no value to bring down.

Example 6: Divide x² + 3 x³ – 5 by 4 + x.

First, arrange both polynomials in descending order, leaving space or filling in a place holder for any missing terms.

Now divide 3 x³ by x and bring this to the top as the first part of the answer.

Multiply 3 x² by the divisor x + 4 and place these in the columns with like terms.

Subtract. Remember, to subtract is to add the opposite.

Bring down the next term and start the procedure again.

Since a value was brought down, start the process over again.

At this point, there is no term to bring down. The –181 is the remainder. As in arithmetic, remainders are written over the divisor. So the final answer to the division problem is

Example 7: Divide 64 x³ – 27 by 4 x – 3.

The answer is 16 x² + 12 x + 9

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