Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are mathematical expressions which comprises of one or more terms, all of which has a variable raised to a power. Dividing polynomials is a crucial function in algebra which involves working out the quotient and remainder as soon as one polynomial is divided by another. In this article, we will explore the different methods of dividing polynomials, consisting of synthetic division and long division, and provide instances of how to apply them.
We will further talk about the importance of dividing polynomials and its applications in multiple domains of math.
Prominence of Dividing Polynomials
Dividing polynomials is an essential operation in algebra that has many uses in diverse domains of mathematics, consisting of number theory, calculus, and abstract algebra. It is used to figure out a broad range of challenges, including figuring out the roots of polynomial equations, calculating limits of functions, and calculating differential equations.
In calculus, dividing polynomials is utilized to find the derivative of a function, which is the rate of change of the function at any point. The quotient rule of differentiation includes dividing two polynomials, that is applied to find the derivative of a function which is the quotient of two polynomials.
In number theory, dividing polynomials is used to learn the features of prime numbers and to factorize large numbers into their prime factors. It is also utilized to study algebraic structures such as rings and fields, that 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 used in various fields of mathematics, involving algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is an approach of dividing polynomials which is utilized to divide a polynomial by a linear factor of the form (x - c), where c is a constant. The method is founded on the fact that if f(x) is a polynomial of degree n, subsequently 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 working out a series of calculations to figure out the remainder and quotient. The answer is a streamlined form of the polynomial which is straightforward to function with.
Long Division
Long division is a method of dividing polynomials that is utilized to divide a polynomial by any other polynomial. The approach is relying on the reality that if f(x) is a polynomial of degree n, and g(x) is a polynomial of degree m, at which point 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 highest degree term of the dividend with the highest degree term of the divisor, and subsequently multiplying the answer by the whole divisor. The answer is subtracted from the dividend to reach the remainder. The method is recurring as far as the degree of the remainder is lower than the degree of the divisor.
Examples of Dividing Polynomials
Here are a number of 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 by the linear factor (x - 1). We could 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. 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 say we want 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:
To start with, we divide the largest degree term of the dividend with the highest degree term of the divisor to get:
6x^2
Subsequently, we multiply the entire divisor with 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)
that simplifies to:
7x^3 - 4x^2 + 9x + 3
We recur the method, dividing the highest degree term of the new dividend, 7x^3, by the largest degree term of the divisor, x^2, to achieve:
7x
Then, we multiply the entire divisor by the quotient term, 7x, to achieve:
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)
that simplifies to:
10x^2 + 2x + 3
We repeat the procedure again, dividing the largest degree term of the new dividend, 10x^2, by the highest degree term of the divisor, x^2, to get:
10
Then, we multiply the whole 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)
which simplifies to:
13x - 10
Thus, the outcome 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 conclusion, dividing polynomials is an essential operation in algebra that has multiple uses in various domains of math. Understanding the various approaches of dividing polynomials, such as synthetic division and long division, could support in working out complicated challenges efficiently. Whether you're a learner struggling to get a grasp algebra or a professional operating in a field that includes polynomial arithmetic, mastering the theories of dividing polynomials is essential.
If you require support comprehending dividing polynomials or any other algebraic theories, think about reaching out to Grade Potential Tutoring. Our adept tutors are available remotely or in-person to give individualized and effective tutoring services to support you succeed. Connect with us today to plan a tutoring session and take your math skills to the next level.
