Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions which consist of one or more terms, each of which has a variable raised to a power. Dividing polynomials is an essential working in algebra that includes finding the quotient and remainder when one polynomial is divided by another. In this blog article, we will investigate the different methods of dividing polynomials, consisting of long division and synthetic division, and give scenarios of how to apply them.
We will further talk about the significance of dividing polynomials and its applications in multiple fields of mathematics.
Importance of Dividing Polynomials
Dividing polynomials is an important function in algebra that has several utilizations in diverse fields of mathematics, including number theory, calculus, and abstract algebra. It is applied to work out a extensive array of challenges, including finding the roots of polynomial equations, working out limits of functions, and working out differential equations.
In calculus, dividing polynomials is used to figure out the derivative of a function, that is the rate of change of the function at any moment. The quotient rule of differentiation includes dividing two polynomials, which is used to figure out the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is utilized to learn the characteristics of prime numbers and to factorize large figures into their prime factors. It is also utilized to study algebraic structures for example rings and fields, which are basic theories in abstract algebra.
In abstract algebra, dividing polynomials is applied to determine polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are applied in many fields of arithmetics, including algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is a method of dividing polynomials that is used to divide a polynomial with a linear factor of the form (x - c), at point which c is a constant. The approach is founded on 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 involves writing the coefficients of the polynomial in a row, using the constant as the divisor, and performing a sequence of workings to find the quotient and remainder. The result is a streamlined form of the polynomial which is simpler to function with.
Long Division
Long division is a method of dividing polynomials which is utilized to divide a polynomial with any other polynomial. The method 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, 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 involves dividing the highest degree term of the dividend with the highest degree term of the divisor, and then multiplying the outcome with the whole divisor. The answer is subtracted of the dividend to obtain the remainder. The procedure is repeated until the degree of the remainder is less 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 could apply 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. Thus, we can express 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 utilize long division to streamline the expression:
To start with, we divide the largest degree term of the dividend by the highest degree term of the divisor to get:
6x^2
Then, we multiply the entire divisor by 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)
which 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 get:
7x
Next, we multiply the whole divisor by the quotient term, 7x, to obtain:
7x^3 - 14x^2 + 7x
We subtract this from 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 recur the method again, dividing the largest 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 total divisor by the quotient term, 10, to obtain:
10x^2 - 20x + 10
We subtract this of the new dividend to obtain the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that simplifies 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
Ultimately, dividing polynomials is a crucial operation in algebra which has several utilized in numerous domains of mathematics. Comprehending the different methods of dividing polynomials, such as synthetic division and long division, can support in solving complicated challenges efficiently. Whether you're a learner struggling to get a grasp algebra or a professional working in a field which consists of polynomial arithmetic, mastering the concept of dividing polynomials is important.
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