Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions which includes one or more 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 methods of dividing polynomials, involving synthetic division and long division, and offer examples of how to use them.
We will also discuss the importance of dividing polynomials and its uses in multiple domains of mathematics.
Prominence of Dividing Polynomials
Dividing polynomials is an essential operation in algebra that has multiple utilizations in many domains of mathematics, involving calculus, number theory, and abstract algebra. It is utilized to figure out a broad spectrum of problems, including figuring out the roots of polynomial equations, figuring out limits of functions, and calculating differential equations.
In calculus, dividing polynomials is used to figure out the derivative of a function, which is the rate of change of the function at any point. The quotient rule of differentiation consists of dividing two polynomials, that is used to find 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 huge numbers into their prime factors. It is further utilized to study algebraic structures for instance rings and fields, that are fundamental ideas in abstract algebra.
In abstract algebra, dividing polynomials is applied to determine polynomial rings, which are algebraic structures which generalize the arithmetic of polynomials. Polynomial rings are used in many domains of mathematics, including 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 technique 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 involves writing the coefficients of the polynomial in a row, using the constant as the divisor, and working out a series of workings to work out the quotient and remainder. The answer is a simplified form of the polynomial that is easier to work with.
Long Division
Long division is a method of dividing polynomials which is utilized to divide a polynomial with another polynomial. The technique 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, 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 involves dividing the greatest degree term of the dividend with the highest degree term of the divisor, and then multiplying the result by the entire divisor. The outcome is subtracted of the dividend to reach the remainder. The procedure is repeated as far as 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 assume we need to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 with 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 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 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 can apply long division to streamline the expression:
To start with, we divide the largest degree term of the dividend by the highest degree term of the divisor to get:
6x^2
Subsequently, we multiply the total divisor with the quotient term, 6x^2, to obtain:
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 streamlines to:
7x^3 - 4x^2 + 9x + 3
We repeat the method, dividing the highest degree term of the new dividend, 7x^3, by the largest degree term of the divisor, x^2, to obtain:
7x
Then, we multiply the total divisor by the quotient term, 7x, to get:
7x^3 - 14x^2 + 7x
We subtract this of the new dividend to get the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
that streamline to:
10x^2 + 2x + 3
We repeat the method again, dividing the largest degree term of the new dividend, 10x^2, with the highest degree term of the divisor, x^2, to get:
10
Next, we multiply the whole divisor by the quotient term, 10, to obtain:
10x^2 - 20x + 10
We subtract this of the new dividend to achieve 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 can express f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Conclusion
In conclusion, dividing polynomials is a crucial operation in algebra which has several applications in multiple domains of mathematics. Getting a grasp of the various approaches of dividing polynomials, such as synthetic division and long division, could support in solving complex challenges efficiently. Whether you're a student struggling to comprehend algebra or a professional working in a domain that consists of polynomial arithmetic, mastering the ideas of dividing polynomials is essential.
If you require help comprehending dividing polynomials or anything related to algebraic theories, think about reaching out to Grade Potential Tutoring. Our experienced teachers are accessible online or in-person to give customized and effective tutoring services to help you succeed. Call us right now to plan a tutoring session and take your math skills to the next level.
