Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions that consist of one or several terms, all of which has a variable raised to a power. Dividing polynomials is an important function in algebra that involves figuring out the quotient and remainder once one polynomial is divided by another. In this blog article, we will examine the various methods of dividing polynomials, consisting of long division and synthetic division, and give instances of how to use them.
We will further discuss the significance of dividing polynomials and its applications in various fields of mathematics.
Prominence of Dividing Polynomials
Dividing polynomials is an important operation in algebra which has several applications in many fields of math, consisting of calculus, number theory, and abstract algebra. It is used to figure out a extensive spectrum of challenges, involving working out the roots of polynomial equations, figuring out limits of functions, and solving differential equations.
In calculus, dividing polynomials is used 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 consists of 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 used to study the features of prime numbers and to factorize large figures into their prime factors. It is further utilized to learn algebraic structures for instance rings and fields, that are basic theories in abstract algebra.
In abstract algebra, dividing polynomials is used to define polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are used in multiple fields of math, comprising of algebraic geometry and algebraic number theory.
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), where c is a constant. The approach is founded on 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 includes writing the coefficients of the polynomial in a row, utilizing the constant as the divisor, and working out a sequence of calculations to work out the remainder and quotient. The outcome is a simplified form of the polynomial which is simpler to work with.
Long Division
Long division is a technique of dividing polynomials which is used to divide a polynomial by any other polynomial. The method 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, where m ≤ n, next the division of f(x) by g(x) provides us 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 with the highest degree term of the divisor, and further multiplying the result with the total divisor. The answer is subtracted of the dividend to reach the remainder. The procedure is repeated until the degree of the remainder is lower compared 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 need to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 with 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 result of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Hence, 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 have 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 streamline the expression:
To start with, we divide the highest degree term of the dividend with the highest degree term of the divisor to attain:
6x^2
Next, we multiply the entire divisor by the quotient term, 6x^2, to obtain:
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)
that simplifies to:
7x^3 - 4x^2 + 9x + 3
We repeat the method, dividing the largest degree term of the new dividend, 7x^3, by the largest degree term of the divisor, x^2, to achieve:
7x
Then, we multiply the entire divisor with the quotient term, 7x, to obtain:
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 simplifies to:
10x^2 + 2x + 3
We repeat the process again, dividing the highest degree term of the new dividend, 10x^2, by the largest degree term of the divisor, x^2, to obtain:
10
Subsequently, we multiply the entire divisor by the quotient term, 10, to obtain:
10x^2 - 20x + 10
We subtract this of the new dividend to get the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that streamlines to:
13x - 10
Hence, the result of the long division is the quotient polynomial 6x^2 - 7x + 9 and the remainder 13x - 10. We could express 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 multiple applications in various fields of mathematics. Understanding the various approaches of dividing polynomials, such as synthetic division and long division, could guide them in figuring out complicated challenges efficiently. Whether you're a student struggling to comprehend algebra or a professional working in a field that includes polynomial arithmetic, mastering the ideas of dividing polynomials is essential.
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