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 an essential function in algebra that involves finding the remainder and quotient once 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 provide scenarios of how to utilize them.
We will further talk about the significance of dividing polynomials and its uses in multiple fields of math.
Importance of Dividing Polynomials
Dividing polynomials is a crucial function in algebra that has multiple applications in many domains of math, involving number theory, calculus, and abstract algebra. It is applied to work out a broad range of challenges, involving figuring out the roots of polynomial equations, figuring out limits of functions, and working out differential equations.
In calculus, dividing polynomials is utilized to work out the derivative of a function, that is the rate of change of the function at any point. The quotient rule of differentiation consists of dividing two polynomials, that is applied to work out the derivative of a function which is the quotient of two polynomials.
In number theory, dividing polynomials is utilized to learn the features of prime numbers and to factorize huge values into their prime factors. It is also used to study algebraic structures for example rings and fields, which are fundamental concepts 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 used in multiple domains of arithmetics, comprising of algebraic geometry and algebraic number theory.
Synthetic Division
Synthetic division is a method 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) gives 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 series of workings to find the quotient and remainder. The answer is a simplified form of the polynomial which is easier to work with.
Long Division
Long division is an approach of dividing polynomials which is used to divide a polynomial with any other polynomial. The method 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) 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 by the highest degree term of the divisor, and then multiplying the answer by the entire divisor. The answer is subtracted of the dividend to get the remainder. The method is recurring until the degree of the remainder is less than 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 by 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 answer 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 assume 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 could use long division to streamline the expression:
First, we divide the largest degree term of the dividend by the largest degree term of the divisor to attain:
6x^2
Then, we multiply the whole divisor with the quotient term, 6x^2, to attain:
6x^4 - 12x^3 + 6x^2
We subtract this from the dividend to obtain 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 recur the process, dividing the largest degree term of the new dividend, 7x^3, by the largest degree term of the divisor, x^2, to get:
7x
Next, we multiply the entire divisor by 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 streamline to:
10x^2 + 2x + 3
We repeat the procedure again, dividing the largest degree term of the new dividend, 10x^2, by the largest degree term of the divisor, x^2, to get:
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 obtain the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that 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 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 which has several applications in multiple domains of mathematics. Understanding the various approaches of dividing polynomials, such as long division and synthetic division, could guide them in solving complex challenges efficiently. Whether you're a student struggling to get a grasp algebra or a professional working in a domain that includes polynomial arithmetic, mastering the theories of dividing polynomials is essential.
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