Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are math expressions that includes one or more terms, each of which has a variable raised to a power. Dividing polynomials is an important function in algebra which includes finding the quotient and remainder when one polynomial is divided by another. In this blog article, we will examine the different approaches of dividing polynomials, involving long division and synthetic division, and give instances of how to apply them.
We will also discuss the significance of dividing polynomials and its utilizations in various domains of math.
Importance of Dividing Polynomials
Dividing polynomials is a crucial function in algebra that has multiple utilizations in various fields of mathematics, including number theory, calculus, and abstract algebra. It is used to work out a broad range of problems, including working out the roots of polynomial equations, calculating limits of functions, and solving 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 time. The quotient rule of differentiation includes dividing two polynomials, that is used to figure out the derivative of a function that is the quotient of two polynomials.
In number theory, dividing polynomials is applied to learn the properties of prime numbers and to factorize huge numbers into their prime factors. It is also utilized to learn algebraic structures for example fields and rings, that are basic concepts in abstract algebra.
In abstract algebra, dividing polynomials is used to specify polynomial rings, that are algebraic structures which generalize the arithmetic of polynomials. Polynomial rings are applied in multiple domains of mathematics, including algebraic geometry and algebraic number theory.
Synthetic Division
Synthetic division is a technique 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 method is founded on 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, utilizing the constant as the divisor, and performing a chain of calculations to work out the remainder and quotient. The answer is a simplified structure of the polynomial that is easier to function with.
Long Division
Long division is a method of dividing polynomials which is utilized to divide a polynomial with another polynomial. The approach 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) gives 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 subsequently multiplying the result by the entire divisor. The answer is subtracted from the dividend to get 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 few 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 could use 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. Thus, 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 with the polynomial g(x) = x^2 - 2x + 1. We can use long division to streamline the expression:
First, we divide the largest degree term of the dividend with the highest degree term of the divisor to get:
6x^2
Then, we multiply the total divisor by the quotient term, 6x^2, to attain:
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 simplifies to:
7x^3 - 4x^2 + 9x + 3
We repeat the process, 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 entire divisor by the quotient term, 7x, to get:
7x^3 - 14x^2 + 7x
We subtract this from the new dividend to obtain 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, with the largest degree term of the divisor, x^2, to achieve:
10
Then, we multiply the whole divisor with the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this of the new dividend to get the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
which simplifies to:
13x - 10
Thus, 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 an important operation in algebra that has many applications in multiple fields of math. Comprehending the various methods of dividing polynomials, such as synthetic division and long division, could help in figuring out intricate problems efficiently. Whether you're a student struggling to understand algebra or a professional working in a domain which involves polynomial arithmetic, mastering the ideas of dividing polynomials is important.
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