Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions which consist of one or more terms, all of which has a variable raised to a power. Dividing polynomials is an important function in algebra which involves figuring out the remainder and quotient when one polynomial is divided by another. In this blog, we will examine the different techniques of dividing polynomials, consisting of long division and synthetic division, and give instances of how to use them.
We will further talk about the importance of dividing polynomials and its utilizations in multiple fields of mathematics.
Significance of Dividing Polynomials
Dividing polynomials is an essential function in algebra which has multiple applications in many domains of mathematics, involving number theory, calculus, and abstract algebra. It is applied to solve a broad range of challenges, involving working out the roots of polynomial equations, figuring out limits of functions, and working out differential equations.
In calculus, dividing polynomials is applied to find the derivative of a function, which is the rate of change of the function at any point. The quotient rule of differentiation consists of dividing two polynomials, which is used to find the derivative of a function which is the quotient of two polynomials.
In number theory, dividing polynomials is used to learn the features of prime numbers and to factorize huge values into their prime factors. It is further applied to study algebraic structures for instance rings and fields, that are rudimental concepts in abstract algebra.
In abstract algebra, dividing polynomials is used to specify polynomial rings, that are algebraic structures which generalize the arithmetic of polynomials. Polynomial rings are used in many domains of mathematics, involving algebraic number theory and algebraic geometry.
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), where 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 consists of writing the coefficients of the polynomial in a row, utilizing the constant as the divisor, and performing a chain of workings to find the remainder and quotient. The result is a streamlined structure of the polynomial which is straightforward to work with.
Long Division
Long division is a technique of dividing polynomials which is applied to divide a polynomial with any other polynomial. The approach is based 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) 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 greatest degree term of the dividend by the highest degree term of the divisor, and then multiplying the answer with the whole divisor. The outcome is subtracted of the dividend to reach the remainder. The procedure is repeated as far as 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 say we want to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We can utilize 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 state f(x) as:
f(x) = (x - 1)(3x^2 + 7x + 2) + 4
Example 2: Long Division
Example 2: Long Division
Let's say we want to divide the polynomial f(x) = 6x^4 - 5x^3 + 2x^2 + 9x + 3 with the polynomial g(x) = x^2 - 2x + 1. We can 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
Then, we multiply the whole divisor with 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 recur the procedure, dividing the largest degree term of the new dividend, 7x^3, with the largest degree term of the divisor, x^2, to obtain:
7x
Subsequently, we multiply the entire divisor with the quotient term, 7x, to achieve:
7x^3 - 14x^2 + 7x
We subtract this of the new dividend to achieve the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
which simplifies to:
10x^2 + 2x + 3
We recur the method again, dividing the highest degree term of the new dividend, 10x^2, by the largest degree term of the divisor, x^2, to get:
10
Subsequently, we multiply the total divisor with the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this from 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 can express f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Conclusion
In conclusion, dividing polynomials is an essential operation in algebra that has multiple uses in numerous fields of mathematics. Understanding the various approaches of dividing polynomials, for instance long division and synthetic division, can help in working out intricate challenges efficiently. Whether you're a student struggling to understand algebra or a professional operating in a field which involves polynomial arithmetic, mastering the theories of dividing polynomials is essential.
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