Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are mathematical expressions that consist of one or several terms, all of which has a variable raised to a power. Dividing polynomials is a crucial operation in algebra that involves finding the remainder and quotient when one polynomial is divided by another. In this blog article, we will explore the different approaches of dividing polynomials, including long division and synthetic division, and offer examples of how to utilize them.
We will also discuss the importance of dividing polynomials and its utilizations in different fields of math.
Significance of Dividing Polynomials
Dividing polynomials is an essential function in algebra which has many utilizations in diverse domains of math, consisting of calculus, number theory, and abstract algebra. It is applied to figure out a wide range of challenges, involving working out the roots of polynomial equations, calculating limits of functions, and calculating differential equations.
In calculus, dividing polynomials is applied to find the derivative of a function, that is the rate of change of the function at any time. The quotient rule of differentiation includes dividing two polynomials, which is applied to work out the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is used to learn the characteristics of prime numbers and to factorize huge values into their prime factors. It is further used to study algebraic structures for example fields and rings, that are rudimental theories in abstract algebra.
In abstract algebra, dividing polynomials is applied to define polynomial rings, that are algebraic structures which generalize the arithmetic of polynomials. Polynomial rings are applied in multiple fields of math, comprising of algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is an approach of dividing polynomials which is utilized to divide a polynomial with a linear factor of the form (x - c), where c is a constant. The technique 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 includes writing the coefficients of the polynomial in a row, using the constant as the divisor, and working out a sequence of workings to work out the remainder and quotient. The result is a simplified structure of the polynomial that is simpler to work 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 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, subsequently 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 by the highest degree term of the divisor, and then multiplying the answer with the entire divisor. The result is subtracted from the dividend to reach the remainder. The method is recurring until the degree of the remainder is less than the degree of the divisor.
Examples of Dividing Polynomials
Here are some examples of dividing polynomial expressions:
Example 1: Synthetic Division
Let's assume 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 simplify 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 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 apply long division to simplify the expression:
First, we divide the highest degree term of the dividend with the largest degree term of the divisor to obtain:
6x^2
Then, 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 streamlines to:
7x^3 - 4x^2 + 9x + 3
We recur the process, dividing the highest degree term of the new dividend, 7x^3, by the largest degree term of the divisor, x^2, to get:
7x
Subsequently, 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 achieve the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
which streamline to:
10x^2 + 2x + 3
We recur the procedure again, dividing the highest degree term of the new dividend, 10x^2, by the highest degree term of the divisor, x^2, to get:
10
Subsequently, we multiply the whole divisor with 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)
which streamlines to:
13x - 10
Hence, the result 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 conclusion, dividing polynomials is a crucial operation in algebra which has multiple applications in numerous fields of mathematics. Comprehending the various techniques of dividing polynomials, such as long division and synthetic division, can support in figuring out intricate problems efficiently. Whether you're a student struggling to understand algebra or a professional operating in a field which consists of polynomial arithmetic, mastering the theories of dividing polynomials is essential.
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