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 function in algebra which includes finding the quotient and remainder when one polynomial is divided by another. In this blog, we will investigate the various approaches of dividing polynomials, consisting of long division and synthetic division, and give examples of how to apply them.
We will further talk about the significance of dividing polynomials and its utilizations in different domains of mathematics.
Importance of Dividing Polynomials
Dividing polynomials is a crucial operation in algebra that has several uses in diverse fields of math, including calculus, number theory, and abstract algebra. It is applied to work out a wide range of challenges, including working out the roots of polynomial equations, working out limits of functions, and calculating differential equations.
In calculus, dividing polynomials is used to work out the derivative of a function, which is the rate of change of the function at any moment. The quotient rule of differentiation consists of dividing two polynomials, that is utilized to work out the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is applied to study the features of prime numbers and to factorize huge values into their prime factors. It is further used to learn algebraic structures such as rings and fields, that are fundamental ideas in abstract algebra.
In abstract algebra, dividing polynomials is applied to specify polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are used in multiple domains of arithmetics, including algebraic geometry and algebraic number theory.
Synthetic Division
Synthetic division is an approach of dividing polynomials which is applied to divide a polynomial with a linear factor of the form (x - c), at point which c is a constant. The technique is based on the fact that if f(x) is a polynomial of degree n, subsequently 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 consists of writing the coefficients of the polynomial in a row, utilizing the constant as the divisor, and carrying out a series of workings to work out the remainder and quotient. The result is a streamlined form of the polynomial that is easier to work with.
Long Division
Long division is a technique of dividing polynomials that is applied to divide a polynomial by another polynomial. The approach is based on the fact that if f(x) is a polynomial of degree n, and g(x) is a polynomial of degree m, where m ≤ n, then 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 involves dividing the greatest degree term of the dividend by the highest degree term of the divisor, and further multiplying the result with the total divisor. The answer is subtracted from the dividend to get the remainder. The process is recurring as far as the degree of the remainder is lower in comparison to 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 assume we have to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 with the linear factor (x - 1). We can apply synthetic division to streamline 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 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 need 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 use long division to simplify the expression:
First, we divide the largest degree term of the dividend by the largest degree term of the divisor to attain:
6x^2
Next, 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 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 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 entire divisor with 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 simplifies to:
10x^2 + 2x + 3
We repeat the procedure again, dividing the largest degree term of the new dividend, 10x^2, by the largest degree term of the divisor, x^2, to achieve:
10
Next, we multiply the entire divisor by the quotient term, 10, to obtain:
10x^2 - 20x + 10
We subtract this from the new dividend to obtain the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that streamlines 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 express f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Conclusion
In Summary, dividing polynomials is an essential operation in algebra that has multiple utilized in multiple fields of math. Getting a grasp of the various methods of dividing polynomials, for instance long division and synthetic division, can guide them in figuring out intricate problems efficiently. Whether you're a student struggling to understand algebra or a professional operating in a field which involves polynomial arithmetic, mastering the ideas of dividing polynomials is essential.
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