Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions that consist of one or more terms, each of which has a variable raised to a power. Dividing polynomials is an important working in algebra which includes finding the quotient and remainder as soon as one polynomial is divided by another. In this blog, we will examine the different approaches of dividing polynomials, including synthetic division and long division, and provide examples of how to use them.
We will further talk about the significance of dividing polynomials and its applications in various fields of math.
Importance of Dividing Polynomials
Dividing polynomials is an essential function in algebra which has several utilizations in various fields of arithmetics, consisting of number theory, calculus, and abstract algebra. It is used to work out a wide spectrum of problems, involving figuring out the roots of polynomial equations, calculating limits of functions, and solving differential equations.
In calculus, dividing polynomials is applied to figure out the derivative of a function, which is the rate of change of the function at any point. The quotient rule of differentiation involves dividing two polynomials, which 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 values into their prime factors. It is further utilized to learn algebraic structures for instance rings and fields, that are rudimental ideas in abstract algebra.
In abstract algebra, dividing polynomials is applied to define polynomial rings, that are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are used in multiple domains of math, involving 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, then 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 includes writing the coefficients of the polynomial in a row, applying the constant as the divisor, and working out a chain of workings to find the quotient and remainder. The result is a streamlined structure of the polynomial which is easier to work with.
Long Division
Long division is an approach of dividing polynomials that is applied to divide a polynomial by any other polynomial. The approach 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, next 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 further multiplying the answer with the whole divisor. The outcome is subtracted of the dividend to reach the remainder. The process is recurring until the degree of the remainder is less in comparison to 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 want to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We can use synthetic division to simplify 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 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 streamline the expression:
First, we divide the largest degree term of the dividend by 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 obtain 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 largest degree term of the new dividend, 7x^3, with the highest degree term of the divisor, x^2, to achieve:
7x
Then, we multiply the whole divisor by the quotient term, 7x, to achieve:
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 streamline to:
10x^2 + 2x + 3
We repeat 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 obtain:
10
Subsequently, 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)
that 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 could state f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Conclusion
Ultimately, dividing polynomials is an essential operation in algebra which has multiple utilized in multiple domains of math. Understanding the different methods of dividing polynomials, for instance synthetic division and long division, could support in working out intricate problems efficiently. Whether you're a student struggling to understand algebra or a professional operating in a field that involves polynomial arithmetic, mastering the concept of dividing polynomials is important.
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