Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions that includes one or several terms, all of which has a variable raised to a power. Dividing polynomials is a crucial function in algebra that involves finding the quotient and remainder as soon as one polynomial is divided by another. In this blog article, we will examine the different techniques of dividing polynomials, involving long division and synthetic division, and give examples of how to use them.
We will also discuss the importance of dividing polynomials and its applications in different fields of mathematics.
Importance of Dividing Polynomials
Dividing polynomials is an essential operation in algebra which has multiple uses in diverse domains of arithmetics, consisting of number theory, calculus, and abstract algebra. It is utilized to figure out a broad spectrum of challenges, consisting of working 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 includes dividing two polynomials, that is used to figure out the derivative of a function which is the quotient of two polynomials.
In number theory, dividing polynomials is utilized to study the properties of prime numbers and to factorize huge numbers into their prime factors. It is further used to study algebraic structures such as rings and fields, that are basic concepts in abstract algebra.
In abstract algebra, dividing polynomials is utilized to define polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are utilized in many domains of math, 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 with a linear factor of the form (x - c), where c is a constant. The method is based on the fact that if f(x) is a polynomial of degree n, subsequently 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 involves writing the coefficients of the polynomial in a row, applying the constant as the divisor, and carrying out a chain of calculations to find the remainder and quotient. The result is a simplified structure of the polynomial which is straightforward to work with.
Long Division
Long division is a method of dividing polynomials which is used to divide a polynomial by any other polynomial. The approach is on the basis 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, then the division of f(x) by g(x) gives a quotient polynomial of degree n-m and a remainder of degree m-1 or less.
The long division algorithm consists of dividing the greatest degree term of the dividend by the highest degree term of the divisor, and then multiplying the outcome with the whole divisor. The answer is subtracted from the dividend to reach the remainder. The procedure is repeated as far as the degree of the remainder is less than the degree of the divisor.
Examples of Dividing Polynomials
Here are some 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 can apply synthetic division to streamline the expression:
1 | 3 4 -5 2 | 3 7 2 |---------- 3 7 2 4
The outcome of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Thus, 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 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 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 repeat the method, dividing the highest degree term of the new dividend, 7x^3, with the highest degree term of the divisor, x^2, to achieve:
7x
Subsequently, we multiply the entire divisor by the quotient term, 7x, to get:
7x^3 - 14x^2 + 7x
We subtract this of the new dividend to obtain the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
which streamline to:
10x^2 + 2x + 3
We recur the procedure again, dividing the highest 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 entire divisor by the quotient term, 10, to obtain:
10x^2 - 20x + 10
We subtract this of the new dividend to obtain the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
which simplifies to:
13x - 10
Hence, the outcome of the long division is the quotient polynomial 6x^2 - 7x + 9 and the remainder 13x - 10. We can state f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Conclusion
Ultimately, dividing polynomials is a crucial operation in algebra that has several uses in numerous fields of mathematics. Understanding the various techniques of dividing polynomials, for example synthetic division and long division, could support in solving complicated problems efficiently. Whether you're a learner struggling to comprehend algebra or a professional working in a field which consists of polynomial arithmetic, mastering the concept of dividing polynomials is essential.
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