Can You Explain How to Divide Two Polynomials? In mathematics, Polynomials are algebraic expressions of constants and variables. The name itself gives us a basic idea of its meaning which is “poly” (meaning many) and “nominal” (meaning terms or parts). In simple […]
Can You Explain How to Divide Two Polynomials?
In mathematics, Polynomials are algebraic expressions of constants and variables. The name itself gives us a basic idea of its meaning which is “poly” (meaning many) and “nominal” (meaning terms or parts). In simple words, polynomials can be defined as a mathematical expression that consists of a variable of non-negative integral power along with their coefficients. As for example, 2 + x, 4z – 13 etc. In these examples, x and z are the variables respectively. The numerical digits included in the expression are called the coefficients.
Polynomials may be used to represent a variety of ordinary mathematical operations. A polynomial can be understood as the sum of the costs of goods in a supermarket bill. A polynomial can be used to calculate the distance traveled by a vehicle or item. Polynomials may be used to calculate the perimeter, area, as well asthe volume of geometric shapes. These are only a few of the many uses for polynomials.
Before going further, let us talk about the degree of a polynomial. The degree of a polynomial can be defined as the highest power of variables in a single term of a polynomial expression. For example,
- 3x + 9, here “x” is the variable. As we can see that the highest power of “x” in this polynomial is 1, hence the degree of this polynomial will be 1.
- 9xz + 4x + 2, In this polynomial, we have two variables. In the first term i.e. “9xz”, the power will be (1 + 1 = 2) and the power in the second term is 1. Thus, the degree of this polynomial will be 2.
- 5xy + 8, in this case, the degrees of the variables in all the terms are (2 + 1 = 3), (! + 1 = 2) respectively. Hence the degree of this polynomial expression will be 3.
The constants in the above expressions have zero power of the variable. These are called constant polynomials. In accordance with the degrees of a polynomial, we can classify them in various types such as,
- Linear polynomial, which is a 1st-degree polynomial.
- Quadratic polynomial, which is a 2nd-degree polynomial.
- Cubic polynomial, which is a 3rd-degree polynomial.
- Biquadratic polynomial, which is a 4th-degree polynomial.
The basic classification of polynomials is based on the number of terms in a polynomial expression. They are-
- Monomial: These polynomial expressions have only one term in them. The power of the variable can be either zero or greater than it. For example, 9; x; 5x; 2z; these are monomials as there is no addition or subtraction exists in between.
- Binomial: These polynomial expressions are represented with the help of two algebraic terms. For example, 9x + 8; 7x + y; 3z + 8 etc.
- Trinomial: Trinomial polynomials are those expressions that consist of three algebraic terms. For example, x + y + z; etc.
- Quadrinomial: These are algebraic expressions that are obtained by adding or subtracting two binomials. This type of polynomials is not used very often.
As we have gone through the various types of polynomials, now it is time to see the operations that can be done between polynomials. Polynomials can be added, subtracted, or multiplied with each other in the same process that we do with the operations of integers. These are very simple methods. Now let us talk about the division of polynomials. There are two methods by which we can divide one polynomial from the other. They are
The synthetic method: which is very simple and can be done very easily if the divider is a monomial. It is done by dividing the coefficients at first and then applying the quotient law to the variables. For example, let us divide 50 by 10x.
First of all, let us divide the coefficients i.e. 50/10 = 5
Now we will have to use the quotient law to divide the variables i.e. = x.
Now multiplying the quotients of both variables and the coefficients will give us the answer i.e. 5 * x = 5x.
Let us have another example such as divide 9 by 3xy.
1st step- 9/3 = 3
2nd step- / xy = = y
3rd step- 3 * y = 3y, the answer is 3y.
More example question and answers: Polynomials From Class 10 Maths
The long division method: The long division method is mostly used method in the division of polynomials. The division process is the same as the division of integers. The steps are explained below;
- Arrange both the divisor and dividend in descending order of variables.
- Divide the 1st term of the dividend by the 1st term of the divisor. This will give us the 1st term of the quotient.
- Multiply all terms of the divisor by the 1st term of the quotient and subtract it from the dividend.
- After that, if we get remainders with greater degrees than the divisor, we will have to do the same method until the remainder is left with lower degrees of a variable than the divisor.