Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are math expressions that consist of one or more terms, all of which has a variable raised to a power. Dividing polynomials is an essential operation in algebra which involves working out the quotient and remainder as soon as one polynomial is divided by another. In this blog article, we will examine the different methods of dividing polynomials, consisting of synthetic division and long division, and provide examples of how to utilize them.
We will further talk about the importance of dividing polynomials and its utilizations in various fields of math.
Importance of Dividing Polynomials
Dividing polynomials is an important function in algebra which has multiple applications in many fields of arithmetics, including calculus, number theory, and abstract algebra. It is utilized to work out a wide array of challenges, involving figuring out the roots of polynomial equations, calculating limits of functions, and solving differential equations.
In calculus, dividing polynomials is applied to work out the derivative of a function, which is the rate of change of the function at any point. The quotient rule of differentiation involves 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 utilized to learn the properties of prime numbers and to factorize huge numbers into their prime factors. It is also used to learn algebraic structures for example fields and rings, that are rudimental concepts 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 fields of arithmetics, including algebraic geometry and algebraic number theory.
Synthetic Division
Synthetic division is an approach of dividing polynomials that is used to divide a polynomial by a linear factor of the form (x - c), at point which c is a constant. The approach 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) 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 workings to work out the quotient and remainder. The outcome is a streamlined form of the polynomial that is straightforward to work with.
Long Division
Long division is a method of dividing polynomials which is applied to divide a polynomial by any other polynomial. The technique is founded 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) offers uf 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 by the highest degree term of the divisor, and further multiplying the result by the whole divisor. The outcome is subtracted of the dividend to obtain the remainder. The procedure 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 say 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 result 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 assume we want 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 simplify the expression:
First, we divide the largest degree term of the dividend with the highest degree term of the divisor to attain:
6x^2
Next, we multiply the whole 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 streamlines to:
7x^3 - 4x^2 + 9x + 3
We recur the procedure, dividing the highest degree term of the new dividend, 7x^3, by the highest degree term of the divisor, x^2, to achieve:
7x
Next, we multiply the total divisor with the quotient term, 7x, to achieve:
7x^3 - 14x^2 + 7x
We subtract this of the new dividend to obtain the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
which simplifies to:
10x^2 + 2x + 3
We repeat the process again, dividing the highest degree term of the new dividend, 10x^2, with the largest degree term of the divisor, x^2, to get:
10
Next, we multiply the entire divisor by the quotient term, 10, to get:
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
Hence, the answer 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 many applications in numerous domains of math. Comprehending the different approaches of dividing polynomials, for instance long division and synthetic division, can support in figuring out complicated challenges efficiently. Whether you're a learner struggling to comprehend algebra or a professional working in a field that consists of polynomial arithmetic, mastering the concept of dividing polynomials is important.
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