Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are mathematical expressions that comprises of one or several terms, each of which has a variable raised to a power. Dividing polynomials is an essential function in algebra which involves working out the quotient and remainder once one polynomial is divided by another. In this blog article, we will explore the various methods of dividing polynomials, consisting of long division and synthetic division, and give scenarios of how to apply them.
We will further discuss the significance of dividing polynomials and its applications in various domains of math.
Importance of Dividing Polynomials
Dividing polynomials is a crucial operation in algebra that has several uses in various fields of mathematics, consisting of calculus, number theory, and abstract algebra. It is applied to figure out a broad array of challenges, consisting of working out the roots of polynomial equations, calculating limits of functions, and calculating differential equations.
In calculus, dividing polynomials is applied to figure out the derivative of a function, that is the rate of change of the function at any point. The quotient rule of differentiation includes 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 utilized to learn the characteristics of prime numbers and to factorize large numbers into their prime factors. It is further utilized to study algebraic structures such as rings and fields, that are basic theories in abstract algebra.
In abstract algebra, dividing polynomials is applied to define polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are utilized in various domains of arithmetics, involving algebraic geometry and algebraic number theory.
Synthetic Division
Synthetic division is a method of dividing polynomials which is used to divide a polynomial with 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, 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, applying the constant as the divisor, and working out a chain of workings to work out the remainder and quotient. The answer is a streamlined structure of the polynomial which is easier to work with.
Long Division
Long division is a method of dividing polynomials that is used to divide a polynomial by another polynomial. The technique 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, subsequently 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 highest degree term of the dividend with the highest degree term of the divisor, and then multiplying the outcome with the entire divisor. The outcome is subtracted from the dividend to get the remainder. The procedure is repeated as far as the degree of the remainder is lower compared to the degree of the divisor.
Examples of Dividing Polynomials
Here are a number of 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 could apply synthetic division to streamline the expression:
1 | 3 4 -5 2 | 3 7 2 |---------- 3 7 2 4
The answer 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 could use 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 whole divisor with the quotient term, 6x^2, to get:
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 largest degree term of the new dividend, 7x^3, by the highest degree term of the divisor, x^2, to get:
7x
Subsequently, we multiply the whole divisor by the quotient term, 7x, to get:
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, with the highest degree term of the divisor, x^2, to obtain:
10
Next, we multiply the whole divisor with 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)
which streamlines to:
13x - 10
Therefore, the result of the long division is the quotient polynomial 6x^2 - 7x + 9 and the remainder 13x - 10. We can state f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Conclusion
Ultimately, dividing polynomials is an important operation in algebra which has many utilized in multiple domains of math. Comprehending the different approaches of dividing polynomials, for example long division and synthetic division, can guide them in figuring out intricate problems efficiently. Whether you're a student struggling to get a grasp algebra or a professional working in a domain which includes polynomial arithmetic, mastering the concept of dividing polynomials is important.
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