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 which includes finding the quotient and remainder when one polynomial is divided by another. In this article, we will explore the various techniques of dividing polynomials, involving long division and synthetic division, and provide examples of how to use them.
We will further talk about the importance of dividing polynomials and its applications in various fields of math.
Significance of Dividing Polynomials
Dividing polynomials is an important operation in algebra that has several uses in many fields of math, including calculus, number theory, and abstract algebra. It is utilized to work out a wide array of problems, consisting of working out the roots of polynomial equations, figuring out limits of functions, and solving differential equations.
In calculus, dividing polynomials is applied to find the derivative of a function, that is the rate of change of the function at any moment. The quotient rule of differentiation involves dividing two polynomials, which is used to find the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is utilized to learn the properties of prime numbers and to factorize large numbers into their prime factors. It is also applied to learn algebraic structures such as rings and fields, which are basic theories in abstract algebra.
In abstract algebra, dividing polynomials is used to determine polynomial rings, which are algebraic structures which generalize the arithmetic of polynomials. Polynomial rings are utilized in multiple domains of arithmetics, involving algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is a method of dividing polynomials which is utilized to divide a polynomial with 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, then 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 carrying out a series of calculations to figure out the quotient and remainder. The outcome is a streamlined form of the polynomial that is easier to work with.
Long Division
Long division is a technique of dividing polynomials that is used to divide a polynomial by another 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) provides us 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 with the highest degree term of the divisor, and subsequently multiplying the outcome with the entire divisor. The result is subtracted from the dividend to obtain the remainder. The process is recurring until the degree of the remainder is lower 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 have to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We could 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. Hence, 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 need to divide the polynomial f(x) = 6x^4 - 5x^3 + 2x^2 + 9x + 3 by the polynomial g(x) = x^2 - 2x + 1. We could apply long division to streamline the expression:
First, we divide the highest degree term of the dividend by the largest degree term of the divisor to obtain:
6x^2
Subsequently, we multiply the total divisor with the quotient term, 6x^2, to obtain:
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 streamlines to:
7x^3 - 4x^2 + 9x + 3
We recur the procedure, dividing the highest 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 total divisor with the quotient term, 7x, to achieve:
7x^3 - 14x^2 + 7x
We subtract this from the new dividend to obtain the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
that streamline to:
10x^2 + 2x + 3
We recur the process again, dividing the highest degree term of the new dividend, 10x^2, with the highest degree term of the divisor, x^2, to achieve:
10
Subsequently, we multiply the total 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)
which simplifies to:
13x - 10
Thus, the answer 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
Ultimately, dividing polynomials is an essential operation in algebra that has many applications in numerous fields of mathematics. Understanding the various methods of dividing polynomials, for instance long division and synthetic division, could support in working 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 theories of dividing polynomials is essential.
If you need help comprehending dividing polynomials or anything related to algebraic concept, consider calling us at Grade Potential Tutoring. Our adept teachers are available online or in-person to provide personalized and effective tutoring services to support you succeed. Contact us today to plan a tutoring session and take your mathematics skills to the next stage.
