Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are mathematical expressions which comprises of one or more terms, each of which has a variable raised to a power. Dividing polynomials is an important function in algebra which involves finding the remainder and quotient as soon as one polynomial is divided by another. In this blog article, we will explore the different methods of dividing polynomials, consisting of long division and synthetic division, and give scenarios of how to utilize them.
We will further talk about the significance of dividing polynomials and its applications in different domains of math.
Significance of Dividing Polynomials
Dividing polynomials is an important operation in algebra which has many uses in diverse fields of math, involving number theory, calculus, and abstract algebra. It is applied to figure out a broad array of challenges, consisting of finding the roots of polynomial equations, working out limits of functions, and working out differential equations.
In calculus, dividing polynomials is applied to figure out 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 utilized to study the properties of prime numbers and to factorize huge figures into their prime factors. It is further used to learn algebraic structures for instance fields and rings, that are fundamental theories in abstract algebra.
In abstract algebra, dividing polynomials is utilized to define polynomial rings, that are algebraic structures which generalize the arithmetic of polynomials. Polynomial rings are applied in many fields of arithmetics, involving algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is a technique of dividing polynomials which is utilized to divide a polynomial with 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, then the division of f(x) by (x - c) offers 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 performing a chain of workings to work out the quotient and remainder. The answer is a streamlined structure of the polynomial which is easier to function with.
Long Division
Long division is an approach of dividing polynomials which is applied to divide a polynomial by another polynomial. The approach is founded on the reality that if f(x) is a polynomial of degree n, and g(x) is a polynomial of degree m, at which point m ≤ n, subsequently 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 highest degree term of the dividend with the highest degree term of the divisor, and then multiplying the result with the total divisor. The outcome is subtracted from the dividend to reach the remainder. The method is recurring until the degree of the remainder is less compared to 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 have to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We could use 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 say 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 utilize 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 obtain:
6x^2
Then, we multiply the whole divisor by 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 streamlines to:
7x^3 - 4x^2 + 9x + 3
We repeat the process, dividing the largest degree term of the new dividend, 7x^3, by the highest degree term of the divisor, x^2, to achieve:
7x
Then, we multiply the entire divisor with the quotient term, 7x, to obtain:
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)
that simplifies to:
10x^2 + 2x + 3
We recur the process again, dividing the largest degree term of the new dividend, 10x^2, by the largest degree term of the divisor, x^2, to obtain:
10
Then, we multiply the whole 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)
which streamlines to:
13x - 10
Therefore, the answer 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 Summary, dividing polynomials is a crucial operation in algebra which has many utilized in various fields of math. Understanding the various approaches of dividing polynomials, for instance long division and synthetic division, could support in solving intricate challenges efficiently. Whether you're a learner struggling to understand algebra or a professional operating in a field that consists of polynomial arithmetic, mastering the theories of dividing polynomials is important.
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