Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are math expressions which comprises of one or more terms, all of which has a variable raised to a power. Dividing polynomials is an essential working in algebra which involves working out the quotient and remainder as soon as one polynomial is divided by another. In this article, we will examine the various techniques of dividing polynomials, involving synthetic division and long division, and offer examples of how to utilize them.
We will further talk about the importance of dividing polynomials and its utilizations in various fields of mathematics.
Significance of Dividing Polynomials
Dividing polynomials is an essential function in algebra that has several applications in many fields of math, involving number theory, calculus, and abstract algebra. It is used to figure out a wide spectrum of problems, including figuring out the roots of polynomial equations, calculating limits of functions, and solving 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 time. The quotient rule of differentiation consists of dividing two polynomials, that is used 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 also applied to study algebraic structures for example rings and fields, which are basic ideas in abstract algebra.
In abstract algebra, dividing polynomials is used to define polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are used in multiple fields of arithmetics, involving algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is a technique of dividing polynomials which is applied 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, therefore 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, applying the constant as the divisor, and working out a sequence of calculations to figure out the quotient and remainder. The result is a simplified structure of the polynomial that is straightforward to work with.
Long Division
Long division is a method of dividing polynomials that is used to divide a polynomial with another polynomial. The technique is founded 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, next 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 with the highest degree term of the divisor, and then multiplying the result by the whole divisor. The answer is subtracted from the dividend to obtain the remainder. The procedure is repeated until the degree of the remainder is lower compared to 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 want to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 with the linear factor (x - 1). We could apply synthetic division to simplify 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 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 with the polynomial g(x) = x^2 - 2x + 1. We could utilize long division to streamline the expression:
To start with, we divide the highest degree term of the dividend by the highest degree term of the divisor to obtain:
6x^2
Then, 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)
that simplifies to:
7x^3 - 4x^2 + 9x + 3
We recur the process, dividing the largest degree term of the new dividend, 7x^3, with the largest degree term of the divisor, x^2, to get:
7x
Next, we multiply the entire 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)
which simplifies to:
10x^2 + 2x + 3
We recur the method again, dividing the largest degree term of the new dividend, 10x^2, with the largest degree term of the divisor, x^2, to obtain:
10
Next, we multiply the whole divisor by the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this from the new dividend to achieve the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that simplifies to:
13x - 10
Therefore, 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 important operation in algebra which has several applications in various fields of mathematics. Getting a grasp of the different techniques of dividing polynomials, for example synthetic division and long division, can support in working out complicated problems efficiently. Whether you're a learner struggling to comprehend algebra or a professional working in a domain that includes polynomial arithmetic, mastering the ideas of dividing polynomials is crucial.
If you desire support comprehending dividing polynomials or any other algebraic concept, consider reaching out to Grade Potential Tutoring. Our adept teachers are available online or in-person to give customized and effective tutoring services to help you succeed. Contact us right now to schedule a tutoring session and take your mathematics skills to the next level.
