# Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples

Polynomials are mathematical expressions which consist of one or more terms, each of which has a variable raised to a power. Dividing polynomials is an essential operation in algebra which involves finding the remainder and quotient once one polynomial is divided by another. In this blog article, we will examine the different methods of dividing polynomials, including synthetic division and long division, and provide examples of how to use them.

We will also discuss the significance of dividing polynomials and its applications in various fields of mathematics.

## Significance of Dividing Polynomials

Dividing polynomials is an important operation in algebra that has multiple uses in various fields of arithmetics, including number theory, calculus, and abstract algebra. It is applied to solve a broad spectrum of problems, involving figuring out the roots of polynomial equations, calculating limits of functions, and working out differential equations.

In calculus, dividing polynomials is applied to work out the derivative of a function, that is the rate of change of the function at any point. The quotient rule of differentiation includes dividing two polynomials, that is applied to work out the derivative of a function that is the quotient of two polynomials.

In number theory, dividing polynomials is applied to learn the properties of prime numbers and to factorize huge numbers into their prime factors. It is further applied to study algebraic structures for example fields and rings, that are basic theories in abstract algebra.

In abstract algebra, dividing polynomials is used to define polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are utilized in various fields of arithmetics, comprising of algebraic geometry and algebraic number theory.

## Synthetic Division

Synthetic division is an approach of dividing polynomials which is used to divide a polynomial by a linear factor of the form (x - c), at point which c is a constant. The method is founded on the fact that if f(x) is a polynomial of degree n, then the division of f(x) by (x - c) offers a quotient polynomial of degree n-1 and a remainder of f(c).

The synthetic division algorithm consists of writing the coefficients of the polynomial in a row, utilizing the constant as the divisor, and performing a chain of calculations to figure out the quotient and remainder. The answer is a simplified form of the polynomial which is straightforward to function with.

## Long Division

Long division is a method of dividing polynomials which is applied to divide a polynomial with another polynomial. The technique is on the basis the fact that if f(x) is a polynomial of degree n, and g(x) is a polynomial of degree m, where m ≤ n, then the division of f(x) by g(x) offers uf a quotient polynomial of degree n-m and a remainder of degree m-1 or less.

The long division algorithm involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and subsequently multiplying the result with the entire divisor. The result is subtracted of the dividend to get the remainder. The procedure is recurring until the degree of the remainder is lower in comparison to the degree of the divisor.

## Examples of Dividing Polynomials

Here are a number of examples of dividing polynomial expressions:

### Example 1: Synthetic Division

Let's say we have to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We could utilize synthetic division to simplify the expression:

1 | 3 4 -5 2 | 3 7 2 |---------- 3 7 2 4

The result of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Hence, we can express f(x) as:

f(x) = (x - 1)(3x^2 + 7x + 2) + 4

### Example 2: Long Division

Example 2: Long Division

Let's assume we want to divide the polynomial f(x) = 6x^4 - 5x^3 + 2x^2 + 9x + 3 by the polynomial g(x) = x^2 - 2x + 1. We can use long division to simplify the expression:

First, we divide the largest degree term of the dividend with the largest degree term of the divisor to attain:

6x^2

Subsequently, we multiply the total divisor with the quotient term, 6x^2, to attain:

6x^4 - 12x^3 + 6x^2

We subtract this from the dividend to obtain the new dividend:

6x^4 - 5x^3 + 2x^2 + 9x + 3 - (6x^4 - 12x^3 + 6x^2)

which simplifies to:

7x^3 - 4x^2 + 9x + 3

We repeat the process, dividing the highest degree term of the new dividend, 7x^3, by the largest degree term of the divisor, x^2, to obtain:

7x

Then, we multiply the total divisor by the quotient term, 7x, to obtain:

7x^3 - 14x^2 + 7x

We subtract this from the new dividend to achieve the new dividend:

7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)

which simplifies to:

10x^2 + 2x + 3

We repeat the procedure again, dividing the largest degree term of the new dividend, 10x^2, with the largest degree term of the divisor, x^2, to achieve:

10

Next, we multiply the entire divisor with the quotient term, 10, to get:

10x^2 - 20x + 10

We subtract this of the new dividend to achieve the remainder:

10x^2 + 2x + 3 - (10x^2 - 20x + 10)

which simplifies to:

13x - 10

Thus, the result of the long division is the quotient polynomial 6x^2 - 7x + 9 and the remainder 13x - 10. We could express f(x) as:

f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)

## Conclusion

In conclusion, dividing polynomials is a crucial operation in algebra that has several applications in numerous fields of mathematics. Getting a grasp of the different methods of dividing polynomials, for instance long division and synthetic division, can support in working out complicated problems efficiently. Whether you're a student struggling to comprehend algebra or a professional working in a field which involves polynomial arithmetic, mastering the ideas of dividing polynomials is crucial.

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