Dividing Polynomials: Definition, Synthetic Division, Long Division, Calculation, Examples

When it comes to polynomial operations, dividing polynomials is an essential skill to master. Dividing polynomials involves dividing one polynomial by another polynomial, which can be done using a variety of methods such as long division, synthetic division, or factoring. In this article, we will explore the definition of dividing polynomials, the different methods used to divide polynomials, and provide step-by-step explanations and examples to help you understand the concept better.

An Introduction to Dividing Polynomials

Before diving into the specifics of dividing polynomials, let’s first understand what a polynomial is. A polynomial is an algebraic expression that consists of variables, coefficients, and exponents. It is written in the form:

aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀

Here, aₙ to a₀ are coefficients, x is the variable, and ⁿ is the highest degree of the polynomial. The degree of a polynomial is determined by the highest exponent of the variable. For example, in the polynomial 2x³ + 5x² - 3x + 1, the degree is 3.

What is Dividing Polynomials?

Dividing polynomials is an arithmetic operation where we divide one polynomial by another polynomial. The dividend is the polynomial being divided, and the divisor is the polynomial dividing the dividend. The result of dividing two polynomials may or may not be a polynomial itself.

The process of dividing polynomials is similar to long division in arithmetic. We divide the terms of the dividend by the terms of the divisor, following certain steps to obtain the quotient and remainder. The division algorithm can be represented as:

Dividend = Divisor * Quotient + Remainder

In polynomial division, we aim to find the quotient polynomial and the remainder polynomial. The quotient represents the result of the division, while the remainder is the leftover terms that cannot be divided further.

Long Division of Polynomials

One method of dividing polynomials is long division. Long division involves dividing each term of the dividend by the first term of the divisor, and then multiplying the divisor by the result to subtract it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor.

Let’s take an example to illustrate the long division of polynomials. Suppose we want to divide 5x² + 3x + 1 by x + 2.

1. Divide the first term of the dividend (5x²) by the first term of the divisor (x) to get 5x.
2. Multiply the divisor (x + 2) by the result (5x) and subtract it from the dividend (5x² + 3x + 1).
   • (x + 2) * 5x = 5x² + 10x
   • Subtracting: (5x² + 3x + 1) - (5x² + 10x) = -7x + 1
3. Bring down the next term from the dividend (-7x) to get -7x + 1.
4. Divide the new expression (-7x + 1) by the first term of the divisor (x) to get -7.
5. Multiply the divisor (x + 2) by the result (-7) and subtract it from the new expression (-7x + 1).
   • (x + 2) * -7 = -7x - 14
   • Subtracting: (-7x + 1) - (-7x - 14) = 15
6. Since the degree of the remainder (15) is less than the degree of the divisor (x + 2), we stop here.

Therefore, the quotient is 5x - 7 and the remainder is 15. The result of dividing 5x² + 3x + 1 by x + 2 is 5x - 7 with a remainder of 15.

Dividing Polynomials by Monomials

When dividing polynomials by monomials (polynomials with a single term), there are two methods that can be used: the splitting the terms method and the factorization method.

Splitting the Terms Method

The splitting the terms method involves separating the terms of the polynomial and simplifying each term individually. Let’s take an example to understand this method better.

Suppose we want to divide 4x² - 6x by 2x.

1. Split the terms of the polynomial: 4x² - 6x becomes (4x²) - (6x).
2. Simplify each term by dividing by the monomial: (4x²) / (2x) - (6x) / (2x).
   • Simplifying: 2x - 3.

Therefore, the quotient is 2x - 3.

Factorization Method

The factorization method involves factoring out a common term from both the numerator and the denominator and simplifying the expression. Let’s take an example to illustrate this method.

Suppose we want to divide 2x³ + 4x² - 6x by 2x.

1. Factor out 2x from both the numerator and the denominator: 2x³ + 4x² - 6x becomes 2x(x² + 2x - 3) / 2x.
2. Cancel out the common term of 2x from the numerator and the denominator: (x² + 2x - 3) / 1.
   • Simplifying: x² + 2x - 3.

Therefore, the quotient is x² + 2x - 3.

Dividing Polynomials by Binomials

Dividing polynomials by binomials (polynomials with two terms) can be done using the long division method. The long division method is similar to the long division of numbers and involves dividing each term of the dividend by the first term of the divisor.

Let’s take an example to understand how to divide polynomials by binomials using the long division method.

Suppose we want to divide 6x³ + 5x² + 4x - 4 by 2x + 1.

1. Divide the first term of the dividend (6x³) by the first term of the divisor (2x) to get 3x².
2. Multiply the divisor (2x + 1) by the result (3x²) and subtract it from the dividend (6x³ + 5x² + 4x - 4).
   • (2x + 1) * 3x² = 6x³ + 3x²
   • Subtracting: (6x³ + 5x² + 4x - 4) - (6x³ + 3x²) = 2x² + 4x - 4
3. Bring down the next term from the dividend (2x²) to get 2x² + 4x - 4.
4. Divide the new expression (2x² + 4x - 4) by the first term of the divisor (2x) to get x.
5. Multiply the divisor (2x + 1) by the result (x) and subtract it from the new expression (2x² + 4x - 4).
   • (2x + 1) * x = 2x² + x
   • Subtracting: (2x² + 4x - 4) - (2x² + x) = 3x - 4
6. Since the degree of the remainder (3x - 4) is less than the degree of the divisor (2x + 1), we stop here.

Therefore, the quotient is 3x² + x and the remainder is 3x - 4. The result of dividing 6x³ + 5x² + 4x - 4 by 2x + 1 is 3x² + x with a remainder of 3x - 4.

Dividing Polynomials Using Synthetic Division

Synthetic division is a method used to divide a polynomial by a linear binomial of the form (x - k). It is a shorter and more efficient alternative to long division, especially when dividing by linear binomials. Synthetic division allows us to divide polynomials quickly by using only the coefficients of the terms, without explicitly writing out the variables.

Let’s take an example to understand how to divide polynomials using synthetic division.

Suppose we want to divide x³ + 2x² - 3x - 10 by x + 2.

1. Write the divisor in the form (x - k) and write the value of k on the left side of the division. Here, the divisor is x + 2, so k is -2.
2. Set up the division by writing the coefficients of the dividend on the right and k on the left.
3. Bring down the coefficient of the highest degree term of the dividend as it is. Here, the leading coefficient is 1 (coefficient of x³).
4. Multiply k with that leading coefficient and write the product below the second coefficient from the left side of the dividend. So, we get -2 * 1 = -2 that we will write below 2.
5. Add the numbers written in the second column. Here, by adding, we get 2 + (-2) = 0.
6. Repeat the same process of multiplication of k with the number obtained in step 5 and write the product in the next column to the right.
7. At last, we will write the final answer, which will be one degree less than the dividend. So, here, in our dividend, the highest degree term is x³, therefore, in the quotient, the highest degree term will be x. Therefore, the answer obtained is x + 0.

Therefore, the quotient is x and the remainder is 0. The result of dividing x³ + 2x² - 3x - 10 by x + 2 is x with no remainder.

Polynomial Division Using Factors

In some cases, we can divide polynomials by factoring out common factors from both the numerator and the denominator. This method simplifies the division process by reducing the degree of the polynomials before dividing. Let’s take an example to illustrate polynomial division using factors.

Suppose we want to divide 6x⁴ - 9x³ + 3x² by 3x².

1. Factor out 3x² from the numerator and the denominator: 6x⁴ - 9x³ + 3x² becomes 3x²(2x² - 3x + 1) / 3x².
2. Cancel out the common factor of 3x² from the numerator and the denominator: (2x² - 3x + 1) / 1.
   • Simplifying: 2x² - 3x + 1.

Therefore, the quotient is 2x² - 3x + 1.

Dividing Polynomials With Remainders

In some cases, when dividing polynomials, we may encounter a remainder that cannot be divided further. The remainder represents the leftover terms after dividing the polynomial. Let’s take an example to understand how to divide polynomials with remainders.

Suppose we want to divide 5x³ + 7x² + 25 by 2x - 3.

1. Divide the first term of the dividend (5x³) by the first term of the divisor (2x) to get 2.5x².
2. Multiply the divisor (2x - 3) by the result (2.5x²) and subtract it from the dividend (5x³ + 7x² + 25).
   • (2x - 3) * 2.5x² = 5x³ - 7.5x²
   • Subtracting: (5x³ + 7x² + 25) - (5x³ - 7.5x²) = 14.5x² + 25
3. Since the degree of the remainder (14.5x² + 25) is equal to or greater than the degree of the divisor (2x - 3), we cannot divide the polynomial further.

Therefore, the quotient is 2.5x² and the remainder is 14.5x² + 25.

Dividing Polynomials with Missing Terms

When dividing polynomials, we may encounter cases where one polynomial has missing terms compared to the other polynomial. In such cases, it is important to be careful and handle the missing terms properly. Let’s take an example to understand how to divide polynomials with missing terms.

Suppose we want to divide 4x³ - 3x by 2x² - 5x + 3.

1. Divide the first term of the dividend (4x³) by the first term of the divisor (2x²) to get 2x.
2. Multiply the divisor (2x² - 5x + 3) by the result (2x) and subtract it from the dividend (4x³ - 3x).
   • (2x² - 5x + 3) * 2x = 4x³ - 10x² + 6x
   • Subtracting: (4x³ - 3x) - (4x³ - 10x² + 6x) = 10x² - 9x
3. Since the dividend does not have a term with x (missing term), we bring it down as 0x to continue the division.
4. Divide the new expression (10x² - 9x) by the first term of the divisor (2x²) to get 5.
5. Multiply the divisor (2x² - 5x + 3) by the result (5) and subtract it from the new expression (10x² - 9x).
   • (2x² - 5x + 3) * 5 = 10x² - 25x + 15
   • Subtracting: (10x² - 9x) - (10x² - 25x + 15) = 16x - 15
6. Since the degree of the remainder (16x - 15) is less than the degree of the divisor (2x² - 5x + 3), we stop here.

Therefore, the quotient is 2x + 5 and the remainder is 16x - 15. The result of dividing 4x³ - 3x by 2x² - 5x + 3 is 2x + 5 with a remainder of 16x - 15.

Polynomial Division Algorithm

The polynomial division algorithm provides a systematic approach to divide polynomials. It involves dividing the terms of the dividend by the terms of the divisor, following certain steps to obtain the quotient and remainder. The polynomial division algorithm can be summarized as follows:

1. Write the dividend and the divisor in standard form, with the terms arranged in descending order of degrees.
2. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
3. Multiply the divisor by the first term of the quotient and subtract it from the dividend to obtain a new polynomial.
4. Bring down the next term from the dividend and repeat steps 2 and 3 until all terms of the dividend have been processed.
5. The final result is the quotient obtained by combining all the terms from step 2, and the remainder is the polynomial obtained in step 4.

By following these steps, we can divide polynomials efficiently and obtain the quotient and remainder.

Types of Polynomials

Polynomials can be classified into several types based on the number of terms they have.

Monomial

A monomial is a polynomial with only one term.

Binomials

A binomial is a polynomial with two terms.

Trinomials

A trinomial is a polynomial with three terms.

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