April 04, 2023

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

Polynomials are math expressions which includes one or several terms, each of which has a variable raised to a power. Dividing polynomials is an important operation in algebra which includes figuring out the quotient and remainder when one polynomial is divided by another. In this blog, we will explore the various techniques of dividing polynomials, consisting of synthetic division and long division, and provide examples of how to use them.


We will further talk about the importance of dividing polynomials and its uses in different fields of mathematics.

Importance of Dividing Polynomials

Dividing polynomials is a crucial operation in algebra which has multiple utilizations in various domains of math, including calculus, number theory, and abstract algebra. It is applied to figure out a wide array of challenges, consisting of figuring out the roots of polynomial equations, figuring out limits of functions, and solving differential equations.


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


In number theory, dividing polynomials is applied to learn the features of prime numbers and to factorize large figures into their prime factors. It is further used to study algebraic structures such as fields and rings, which are rudimental concepts in abstract algebra.


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

Synthetic Division

Synthetic division is a technique of dividing polynomials which is applied to divide a polynomial with a linear factor of the form (x - c), at point which c is a constant. The approach is founded on the fact that if f(x) is a polynomial of degree n, subsequently the division of f(x) by (x - c) provides 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, applying the constant as the divisor, and working out a series of workings to figure out the quotient and remainder. The outcome is a simplified structure of the polynomial that is simpler to function with.

Long Division

Long division is a method of dividing polynomials which is applied to divide a polynomial by another polynomial. The method is relying on 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) provides us a quotient polynomial of degree n-m and a remainder of degree m-1 or less.


The long division algorithm includes dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the outcome with the entire divisor. The answer is subtracted from the dividend to get the remainder. The procedure is recurring as far as 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 need to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We could use synthetic division to simplify the expression:


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


The answer of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Thus, 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 say we have 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 apply long division to simplify the expression:


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


6x^2


Subsequently, we multiply the whole divisor by the quotient term, 6x^2, to get:


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


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


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


that streamlines to:


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


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


7x


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


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


We subtract this of the new dividend to get the new dividend:


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


which simplifies to:


10x^2 + 2x + 3


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


10


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


10x^2 - 20x + 10


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


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


which streamlines 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 state f(x) as:


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

Conclusion

In Summary, dividing polynomials is an important operation in algebra that has several uses in multiple fields of mathematics. Understanding the different approaches of dividing polynomials, for instance long division and synthetic division, could support in figuring out complex problems efficiently. Whether you're a student struggling to comprehend algebra or a professional working in a field that involves polynomial arithmetic, mastering the theories of dividing polynomials is essential.


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