Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are math expressions that comprises of one or more terms, each of which has a variable raised to a power. Dividing polynomials is an essential working in algebra that involves finding the quotient and remainder once one polynomial is divided by another. In this blog, we will examine the different approaches of dividing polynomials, including synthetic division and long division, and give examples of how to utilize them.
We will further discuss the importance of dividing polynomials and its uses in different fields of mathematics.
Importance of Dividing Polynomials
Dividing polynomials is an important operation in algebra that has multiple applications in various fields of mathematics, consisting of number theory, calculus, and abstract algebra. It is applied to solve a broad array of problems, including working out the roots of polynomial equations, calculating limits of functions, and calculating differential equations.
In calculus, dividing polynomials is applied to figure out the derivative of a function, which is the rate of change of the function at any time. The quotient rule of differentiation involves dividing two polynomials, that is utilized to work out 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 values into their prime factors. It is further used to study algebraic structures for example rings and fields, that are basic theories in abstract algebra.
In abstract algebra, dividing polynomials is used to define polynomial rings, that are algebraic structures which generalize the arithmetic of polynomials. Polynomial rings are applied in many fields of mathematics, including algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is a technique of dividing polynomials that is utilized to divide a polynomial with a linear factor of the form (x - c), where c is a constant. The method 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) 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, applying the constant as the divisor, and performing a series of workings to figure out the remainder and quotient. The answer is a streamlined form of the polynomial which is straightforward to work with.
Long Division
Long division is an approach of dividing polynomials which is applied to divide a polynomial by any other polynomial. The method is on the basis the fact that if f(x) is a polynomial of degree n, and g(x) is a polynomial of degree m, where m ≤ n, next 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 highest degree term of the dividend by the highest degree term of the divisor, and further multiplying the outcome with the whole divisor. The outcome is subtracted of the dividend to reach the remainder. The procedure is recurring until the degree of the remainder is less in comparison to the degree of the divisor.
Examples of Dividing Polynomials
Here are few examples of dividing polynomial expressions:
Example 1: Synthetic Division
Let's say we need to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by 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 answer of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Hence, 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 have 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 use long division to streamline the expression:
First, we divide the highest 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 attain:
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 repeat the procedure, dividing the largest 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 whole divisor by the quotient term, 7x, to achieve:
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)
that simplifies to:
10x^2 + 2x + 3
We repeat the process again, dividing the highest degree term of the new dividend, 10x^2, by the highest degree term of the divisor, x^2, to achieve:
10
Next, we multiply the whole divisor by the quotient term, 10, to obtain:
10x^2 - 20x + 10
We subtract this from the new dividend to achieve the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that simplifies 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 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 many applications in multiple domains of mathematics. Getting a grasp of the various methods of dividing polynomials, such as synthetic division and long division, can support in figuring out complicated challenges efficiently. Whether you're a student struggling to comprehend algebra or a professional operating in a field that involves polynomial arithmetic, mastering the theories of dividing polynomials is important.
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