Dividing Polynomials by Binomials

Introduction to Polynomial Division

Dividing polynomials is a fundamental skill in algebra, especially when simplifying expressions or solving equations. While division by a monomial is straightforward, division by a binomial or any polynomial with more than one term requires a systematic approach known as long division.

• Monomial Division: Each term of the polynomial is divided separately by the monomial.

• Binomial/Polynomial Division: Requires the long division process, similar to dividing whole numbers.

Key Terms in Polynomial Division

• Dividend: The polynomial being divided.

• Divisor: The polynomial by which you are dividing.

• Quotient: The result of the division.

• Remainder: Any leftover part after division.

Long Division of Polynomials

Steps for Dividing a Polynomial by a Binomial

The process of long division for polynomials mirrors the steps used in numerical long division. The procedure is repetitive and follows four main steps:

1. Divide: Divide the first term of the dividend by the first term of the divisor.

2. Multiply: Multiply the entire divisor by the result from step 1 and align like terms under the dividend.

3. Subtract: Subtract the product from the dividend, changing the sign of each term in the lower expression and then adding.

4. Bring Down: Bring down the next term from the original dividend and repeat the process until the degree of the remainder is less than the degree of the divisor.

Arranging Terms

• Arrange both the dividend and divisor in descending powers of x.

• If terms are missing in the dividend, insert them with a coefficient of 0 as placeholders to keep terms aligned.

Examples of Polynomial Long Division

Example 1: Division with No Remainder

Divide by .

1. Arrange terms: Both polynomials are already in descending order.

2. Divide by : Result is .

3. Multiply by : .

4. Subtract: .

5. Bring down next term: .

6. Divide by : Result is $3$.

7. Multiply by $3.

8. Subtract: .

Quotient: Remainder: $0$

Example 2: Division with Placeholders

Divide by .

• Insert missing terms with zero coefficients to align like terms.

• Proceed with long division as above, dividing, multiplying, subtracting, and bringing down terms.

• Stop when the degree of the remainder is less than the degree of the divisor.

Quotient: (Result from division) Remainder: (If any, write as a fraction over the divisor)

Example 3: Division with Multiple Steps

Divide by .

1. Arrange terms in descending order.

2. Divide by : .

3. Multiply by : .

4. Subtract: .

5. Divide by : $5$.

6. Multiply by $5.

7. Subtract: .

8. Since $30\frac{30}{x+3}$.

Quotient: Remainder: $30$ Final Answer:

General Format for Answers

When the division is complete, the answer should be written as:

• If remainder is zero: Quotient only

• If remainder is not zero: Quotient + (Remainder/Divisor)

Summary Table: Steps in Polynomial Long Division

Key Points to Remember

• Always arrange polynomials in descending order of degree.

• Use zero coefficients for missing terms to keep alignment.

• Write the final answer as Quotient + (Remainder/Divisor) if there is a remainder.

• Long division is essential for simplifying rational expressions and solving higher-degree equations.

Additional info:

• Polynomial long division is foundational for later topics such as synthetic division and rational expressions.

• Understanding the process helps in factoring and solving polynomial equations.