Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions which consist of one or more terms, each of which has a variable raised to a power. Dividing polynomials is a crucial operation in algebra that includes figuring out the remainder and quotient as soon as one polynomial is divided by another. In this blog, we will examine the different techniques of dividing polynomials, including long division and synthetic division, and offer examples of how to utilize them.
We will also discuss the importance of dividing polynomials and its utilizations in multiple domains of mathematics.
Importance of Dividing Polynomials
Dividing polynomials is a crucial function in algebra which has multiple utilizations in diverse fields of math, involving number theory, calculus, and abstract algebra. It is utilized to work out a wide array of problems, including finding the roots of polynomial equations, figuring out limits of functions, and solving differential equations.
In calculus, dividing polynomials is applied to work out the derivative of a function, that is the rate of change of the function at any time. The quotient rule of differentiation involves dividing two polynomials, that is used to work out the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is applied to study the characteristics of prime numbers and to factorize large figures into their prime factors. It is further applied to learn algebraic structures for instance rings and fields, that are fundamental ideas in abstract algebra.
In abstract algebra, dividing polynomials is used to define polynomial rings, which are algebraic structures which generalize the arithmetic of polynomials. Polynomial rings are applied in various fields of math, involving algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is a technique of dividing polynomials which is used to divide a polynomial by a linear factor of the form (x - c), where c is a constant. The method is on the basis of the fact that if f(x) is a polynomial of degree n, then the division of f(x) by (x - c) offers 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, using the constant as the divisor, and performing a sequence of workings to figure out the quotient and remainder. The outcome is a simplified structure of the polynomial which is straightforward to work with.
Long Division
Long division is a method of dividing polynomials which is utilized to divide a polynomial by another 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, 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 includes dividing the greatest degree term of the dividend by the highest degree term of the divisor, and then multiplying the outcome by the total divisor. The answer is subtracted of the dividend to obtain the remainder. The procedure is recurring 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 assume we want to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We can use 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. 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 streamline the expression:
First, we divide the largest degree term of the dividend by the highest degree term of the divisor to get:
6x^2
Next, we multiply the total divisor by 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 highest degree term of the new dividend, 7x^3, by the highest degree term of the divisor, x^2, to achieve:
7x
Next, we multiply the whole divisor by the quotient term, 7x, to get:
7x^3 - 14x^2 + 7x
We subtract this of 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 recur the process 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
Next, we multiply the entire divisor by the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this from the new dividend to obtain the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that 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 could state f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Conclusion
In Summary, dividing polynomials is an important operation in algebra that has several applications in multiple domains of mathematics. Understanding the various approaches of dividing polynomials, for example long division and synthetic division, could help in figuring out complicated challenges efficiently. Whether you're a learner struggling to understand algebra or a professional operating in a domain which includes polynomial arithmetic, mastering the concept of dividing polynomials is important.
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