Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions that comprises of one or several terms, each of which has a variable raised to a power. Dividing polynomials is a crucial function in algebra that involves finding the remainder and quotient when one polynomial is divided by another. In this blog article, we will investigate the various methods of dividing polynomials, including synthetic division and long division, and provide examples of how to utilize them.
We will also talk about the significance of dividing polynomials and its utilizations in multiple domains of math.
Prominence of Dividing Polynomials
Dividing polynomials is an essential function in algebra which has several applications in diverse fields of math, consisting of calculus, number theory, and abstract algebra. It is used to figure out a wide range of problems, consisting of figuring out the roots of polynomial equations, calculating limits of functions, and solving 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 involves dividing two polynomials, which is used to figure out the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is used to learn the properties of prime numbers and to factorize large numbers into their prime factors. It is also used to learn algebraic structures for instance rings and fields, which are rudimental theories in abstract algebra.
In abstract algebra, dividing polynomials is applied to determine polynomial rings, that are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are utilized in multiple domains of math, 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), where c is a constant. The technique is based on the fact that if f(x) is a polynomial of degree n, then 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 sequence of calculations to work out the quotient and remainder. The answer is a simplified structure of the polynomial that is easier to function with.
Long Division
Long division is an approach of dividing polynomials that is used to divide a polynomial with any other polynomial. The approach is on the basis 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, next the division of f(x) by g(x) provides us a quotient polynomial of degree n-m and a remainder of degree m-1 or less.
The long division algorithm consists of dividing the greatest degree term of the dividend with the highest degree term of the divisor, and then multiplying the answer with the whole divisor. The outcome is subtracted of the dividend to obtain the remainder. The process is recurring as far as the degree of the remainder is lower compared to the degree of the divisor.
Examples of Dividing Polynomials
Here are some 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 with 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 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 assume 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 can apply long division to streamline the expression:
To start with, we divide the highest degree term of the dividend with the highest degree term of the divisor to attain:
6x^2
Subsequently, 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 repeat the process, dividing the highest degree term of the new dividend, 7x^3, with the largest degree term of the divisor, x^2, to get:
7x
Then, we multiply the total divisor by the quotient term, 7x, to obtain:
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 streamline to:
10x^2 + 2x + 3
We repeat the process again, dividing the highest 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 whole divisor by the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this of the new dividend to achieve the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
which streamlines to:
13x - 10
Therefore, the outcome 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 that has many utilized in various domains of math. Comprehending the different approaches of dividing polynomials, for instance long division and synthetic division, could support in solving complex problems efficiently. Whether you're a student struggling to understand algebra or a professional operating in a domain which involves polynomial arithmetic, mastering the concept of dividing polynomials is important.
If you need support comprehending dividing polynomials or anything related to algebraic theories, consider reaching out to Grade Potential Tutoring. Our experienced tutors are accessible online or in-person to provide customized and effective tutoring services to support you succeed. Contact us today to schedule a tutoring session and take your math skills to the next stage.
