Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are math expressions which consist of one or several terms, all of which has a variable raised to a power. Dividing polynomials is an essential working in algebra that includes finding the remainder and quotient as soon as one polynomial is divided by another. In this article, we will explore the different approaches of dividing polynomials, consisting of long division and synthetic division, and provide instances of how to utilize them.
We will further talk about the significance of dividing polynomials and its uses in different domains of mathematics.
Prominence of Dividing Polynomials
Dividing polynomials is a crucial operation in algebra that has several uses in various domains of mathematics, including number theory, calculus, and abstract algebra. It is applied to solve a broad spectrum of problems, including figuring out the roots of polynomial equations, calculating limits of functions, and solving differential equations.
In calculus, dividing polynomials is used to work out the derivative of a function, that is the rate of change of the function at any moment. The quotient rule of differentiation includes 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 applied to learn the features of prime numbers and to factorize large numbers into their prime factors. It is further applied to learn algebraic structures for example rings and fields, which are basic concepts in abstract algebra.
In abstract algebra, dividing polynomials is used to specify polynomial rings, that are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are utilized in various domains of mathematics, involving algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is an approach of dividing polynomials that is applied to divide a polynomial by a linear factor of the form (x - c), at point which c is a constant. The method is founded on the fact that if f(x) is a polynomial of degree n, therefore 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 consists of writing the coefficients of the polynomial in a row, applying the constant as the divisor, and carrying out a sequence of workings to work out the quotient and remainder. The answer is a streamlined form of the polynomial which is easier to function with.
Long Division
Long division is a technique of dividing polynomials which is utilized to divide a polynomial by another polynomial. The method is relying on the fact 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) 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 greatest degree term of the dividend with the highest degree term of the divisor, and subsequently multiplying the result by the whole divisor. The answer is subtracted of the dividend to obtain the remainder. The process is recurring until the degree of the remainder is lower than 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 need to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We could utilize synthetic division to streamline the expression:
1 | 3 4 -5 2 | 3 7 2 |---------- 3 7 2 4
The result of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Therefore, 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 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 apply long division to streamline the expression:
First, we divide the highest degree term of the dividend with the highest degree term of the divisor to get:
6x^2
Then, we multiply the whole divisor by the quotient term, 6x^2, to get:
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)
which streamlines to:
7x^3 - 4x^2 + 9x + 3
We recur the procedure, dividing the largest degree term of the new dividend, 7x^3, with the highest degree term of the divisor, x^2, to get:
7x
Subsequently, we multiply the entire divisor with 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)
that simplifies to:
10x^2 + 2x + 3
We recur the procedure again, dividing the largest 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 whole divisor with 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)
which simplifies to:
13x - 10
Hence, the answer 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
Ultimately, dividing polynomials is an essential operation in algebra that has several utilized in numerous fields of mathematics. Getting a grasp of the various approaches of dividing polynomials, for instance long division and synthetic division, could support in solving complicated challenges efficiently. Whether you're a learner struggling to understand algebra or a professional operating in a field that includes polynomial arithmetic, mastering the ideas of dividing polynomials is essential.
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