Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions that consist of one or several terms, each of which has a variable raised to a power. Dividing polynomials is an important working in algebra that involves finding the quotient and remainder once one polynomial is divided by another. In this blog, we will explore the different approaches of dividing polynomials, consisting of long division and synthetic division, and offer examples of how to use them.
We will further talk about the importance of dividing polynomials and its applications in multiple domains of math.
Significance of Dividing Polynomials
Dividing polynomials is an important operation in algebra which has several utilizations in various domains of mathematics, involving calculus, number theory, and abstract algebra. It is used to work out a wide range of challenges, consisting of finding the roots of polynomial equations, calculating limits of functions, and calculating differential equations.
In calculus, dividing polynomials is applied to work out the derivative of a function, which is the rate of change of the function at any moment. The quotient rule of differentiation includes dividing two polynomials, which is utilized 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 characteristics of prime numbers and to factorize huge figures into their prime factors. It is also applied to study algebraic structures for instance rings and fields, that are fundamental concepts in abstract algebra.
In abstract algebra, dividing polynomials is applied to define polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are applied in various domains of mathematics, comprising of algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is a technique of dividing polynomials that 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 based on the fact that if f(x) is a polynomial of degree n, therefore the division of f(x) by (x - c) gives 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, using the constant as the divisor, and carrying out a chain of workings to work out the remainder and quotient. The result is a simplified structure of the polynomial that is straightforward to function with.
Long Division
Long division is an approach of dividing polynomials which is used to divide a polynomial by another polynomial. The approach is relying on the reality that if f(x) is a polynomial of degree n, and g(x) is a polynomial of degree m, at which point m ≤ n, subsequently 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 consists of dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the answer with the total divisor. The outcome is subtracted of the dividend to reach the remainder. The method is repeated until the degree of the remainder is lower than 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 assume we want to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We could use synthetic division to streamline 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. Hence, we can state f(x) as:
f(x) = (x - 1)(3x^2 + 7x + 2) + 4
Example 2: Long Division
Example 2: Long Division
Let's assume we need to divide the polynomial f(x) = 6x^4 - 5x^3 + 2x^2 + 9x + 3 with the polynomial g(x) = x^2 - 2x + 1. We could use long division to simplify the expression:
First, we divide the largest degree term of the dividend by the highest degree term of the divisor to get:
6x^2
Next, we multiply the entire divisor by the quotient term, 6x^2, to attain:
6x^4 - 12x^3 + 6x^2
We subtract this from the dividend to get the new dividend:
6x^4 - 5x^3 + 2x^2 + 9x + 3 - (6x^4 - 12x^3 + 6x^2)
that simplifies to:
7x^3 - 4x^2 + 9x + 3
We recur 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
Subsequently, we multiply the entire divisor by the quotient term, 7x, to achieve:
7x^3 - 14x^2 + 7x
We subtract this from the new dividend to get the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
that simplifies to:
10x^2 + 2x + 3
We repeat the method again, dividing the highest degree term of the new dividend, 10x^2, by the highest degree term of the divisor, x^2, to achieve:
10
Subsequently, 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
Therefore, the answer of the long division is the quotient polynomial 6x^2 - 7x + 9 and the remainder 13x - 10. We can express 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 several uses in numerous fields of mathematics. Understanding the various approaches of dividing polynomials, such as synthetic division and long division, could help in working out intricate problems efficiently. Whether you're a learner struggling to understand algebra or a professional working in a domain which includes polynomial arithmetic, mastering the ideas of dividing polynomials is crucial.
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