A polynomial is an algebraic expression consisting of variables and constants. The word polynomial is the combination of two terms i.e., ‘poly’ means ‘many’ and ‘nominal’ means ‘terms’. Collectively, it is said as ‘many terms’.
The general representation of a polynomial is given by:
P(x) = anxn + an-1xn-1+…………………+a2x2 + a1x + a0
Here, n is any positive integer. an,an-1,…,a1,a0 are the coefficients of variable x from the highest degree to the lowest. The highest power of x is the degree of the polynomial.
For example, x2-2x+2 is a polynomial where 1,-2 and 2 are the coefficients and degree is 2.
Division of Polynomials
We can perform operations such as addition, subtraction, multiplication and division on polynomials. Here we will learn dividing a polynomial by another polynomial, through different methods. Also, there are different types of polynomial division, such as:
- Monomial by monomial
- Polynomial by monomial
- Polynomial by polynomial
Let us see examples of each of the above.
Monomial Divided By Monomial
Dividing a monomial by another monomial is very easy because monomial is a single term. Suppose 14x2 is a monomial which is divided by another monomial say 2x, then we get:
14x2 ÷ 2x = (2 × 7 × x × x) ÷ (2 × x)
Cancel the common terms, i.e. 2x.
14x2 ÷ 2x = 7x
Polynomial Divided by Monomial
When a polynomial is divided by a monomial, then each term of the given polynomial is divided by the monomial. Let us consider, P(x) 3x2-x+1 is a polynomial divided by 3x, then;
= 3x2/3x – x/3x + 1/3x
Polynomial Divided by Polynomial
There are two methods by which we can divide a polynomial by another polynomial. They are: Long division method and Synthetic division.
The long division method is very much similar to the basic division method, which we perform on integers. Here the first polynomial is the dividend and another polynomial is the divisor. We divide the dividend with the multiples of the divisor. If any remainder is left, then again we perform the division method. See the example below:
In case of synthetic division, the degree of the divisor polynomial should be equal to 1. This is a shortcut method that takes less time than long division. Thus, we find here the zeros of polynomials.
For example, y=x2+5x+6 is a polynomial. We can factor this polynomial as:
y = (x + 3)(x + 2)
Hence, we can easily find the zeros of polynomial such as:
x=-3 and x=-2
Hence, we have learned so far how to divide the polynomials using different methods. Now we can try solving questions based on them.