Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions which includes one or several terms, each of which has a variable raised to a power. Dividing polynomials is an important working in algebra which includes figuring out the quotient and remainder when one polynomial is divided by another. In this article, we will examine the different approaches of dividing polynomials, consisting of synthetic division and long division, and provide examples of how to use them.
We will also talk about the importance of dividing polynomials and its applications in different domains of mathematics.
Importance of Dividing Polynomials
Dividing polynomials is an essential function in algebra that has several applications in diverse domains of mathematics, consisting of calculus, number theory, and abstract algebra. It is applied to figure out a extensive range of challenges, including working out the roots of polynomial equations, figuring out limits of functions, and calculating differential equations.
In calculus, dividing polynomials is utilized to find the derivative of a function, that is the rate of change of the function at any moment. The quotient rule of differentiation consists of dividing two polynomials, that is utilized to figure out the derivative of a function which 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 also applied to study algebraic structures such as fields and rings, that are rudimental concepts in abstract algebra.
In abstract algebra, dividing polynomials is used to determine polynomial rings, that are algebraic structures which generalize the arithmetic of polynomials. Polynomial rings are applied in various fields of math, including algebraic geometry and algebraic number theory.
Synthetic Division
Synthetic division is an approach of dividing polynomials that is applied to divide a polynomial with a linear factor of the form (x - c), where c is a constant. The approach 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, utilizing the constant as the divisor, and performing a sequence of workings to work out the remainder and quotient. The outcome is a simplified form of the polynomial that is simpler to function with.
Long Division
Long division is a technique of dividing polynomials which is used to divide a polynomial by another polynomial. The technique is based on the reality that if f(x) is a polynomial of degree n, and g(x) is a polynomial of degree m, where 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 includes dividing the highest degree term of the dividend by the highest degree term of the divisor, and subsequently multiplying the answer by the entire divisor. The outcome is subtracted of the dividend to reach the remainder. The method is repeated until the degree of the remainder is less 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 say we have to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 with the linear factor (x - 1). We can apply 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 say we have 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 apply long division to streamline the expression:
First, we divide the highest degree term of the dividend by the highest degree term of the divisor to obtain:
6x^2
Subsequently, we multiply the entire divisor with the quotient term, 6x^2, to attain:
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)
which streamlines to:
7x^3 - 4x^2 + 9x + 3
We repeat the method, dividing the highest degree term of the new dividend, 7x^3, with the largest degree term of the divisor, x^2, to obtain:
7x
Then, we multiply the entire divisor with 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 largest degree term of the new dividend, 10x^2, by the largest degree term of the divisor, x^2, to achieve:
10
Subsequently, we multiply the entire divisor with 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)
that simplifies to:
13x - 10
Thus, the outcome 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
In Summary, dividing polynomials is an essential operation in algebra that has many utilized in multiple domains of mathematics. Understanding the various approaches of dividing polynomials, for example synthetic division and long division, could guide them in figuring out complex challenges efficiently. Whether you're a student struggling to get a grasp algebra or a professional working in a domain that consists of polynomial arithmetic, mastering the theories of dividing polynomials is important.
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