Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions that comprises of one or several terms, each of which has a variable raised to a power. Dividing polynomials is an important working in algebra which involves finding the remainder and quotient once one polynomial is divided by another. In this blog, we will examine the different methods of dividing polynomials, including long division and synthetic division, and offer scenarios of how to use them.
We will further discuss the importance of dividing polynomials and its utilizations in various fields of math.
Importance of Dividing Polynomials
Dividing polynomials is an important operation in algebra which has many uses in many domains of math, consisting of number theory, calculus, and abstract algebra. It is utilized to solve a extensive array of challenges, including figuring out the roots of polynomial equations, figuring out limits of functions, and calculating 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 consists of dividing two polynomials, which is applied to find the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is utilized to study the properties of prime numbers and to factorize large numbers into their prime factors. It is also applied to study algebraic structures for instance rings and fields, that are rudimental concepts in abstract algebra.
In abstract algebra, dividing polynomials is utilized to specify polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are utilized in multiple fields of arithmetics, including algebraic geometry and algebraic number theory.
Synthetic Division
Synthetic division is an approach 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 on the basis of the fact that if f(x) is a polynomial of degree n, therefore 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 involves writing the coefficients of the polynomial in a row, applying the constant as the divisor, and performing a sequence of workings to work out the quotient and remainder. The answer is a simplified structure of the polynomial which is easier to work with.
Long Division
Long division is a technique of dividing polynomials that is applied to divide a polynomial by any other polynomial. The method 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, next 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 includes dividing the greatest 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 from the dividend to get the remainder. The process is recurring as far as the degree of the remainder is lower than the degree of the divisor.
Examples of Dividing Polynomials
Here are few examples of dividing polynomial expressions:
Example 1: Synthetic Division
Let's assume we need to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We can apply synthetic division to streamline 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. Therefore, 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 need to divide the polynomial f(x) = 6x^4 - 5x^3 + 2x^2 + 9x + 3 by the polynomial g(x) = x^2 - 2x + 1. We could use long division to simplify the expression:
First, we divide the highest degree term of the dividend with the largest degree term of the divisor to attain:
6x^2
Next, we multiply the entire divisor with 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)
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 highest degree term of the divisor, x^2, to achieve:
7x
Next, we multiply the whole 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)
that simplifies to:
10x^2 + 2x + 3
We recur the process again, dividing the largest degree term of the new dividend, 10x^2, by the highest degree term of the divisor, x^2, to get:
10
Next, we multiply the total divisor by the quotient term, 10, to obtain:
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 state f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Conclusion
Ultimately, dividing polynomials is an important operation in algebra which has multiple applications in various domains of math. Understanding the different methods of dividing polynomials, for instance long division and synthetic division, can help in figuring out complicated problems efficiently. Whether you're a student struggling to comprehend algebra or a professional working in a domain that consists of polynomial arithmetic, mastering the concept of dividing polynomials is essential.
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