What Are Polynomials?
Polynomial is an algebraic expression that consists of coefficients and variables. Arithmetic operations such as addition, multiplication, subtraction, and positive integer exponentiation of variables are included under polynomials. For example, y2 – 9y + 20 is a polynomial. y is referred to as the variable, 1 and 9 are known as coefficients of the variable, 20 is the constant term, and basic arithmetic operations are used to create the expression. In this article, we will learn more about the various concepts associated with polynomials.
Classification of Polynomials
- Monomial – have only one term in the expression. E.g., 7m2, 8xv
- Binomial – have only two terms or two monomials in the algebraic expression. E.g. 7m2 + 8xv
- Trinomial – have only three terms in the expression. Eg 7m2 + 8xv – 9
Degree of Polynomials
The highest exponent that occurs within a polynomial is known as the degree of that polynomial. In other words, the highest degree of a monomial within the polynomial gives the degree. For example, 7y2 + y – 14 is a polynomial of degree 2 as the highest exponent is 2.
In algebra, the algorithm for dividing a polynomial by another polynomial that is of the same or lower degree is known as polynomial long division or just long division. Let us take a look at the steps that are required for dividing polynomials. Suppose we have to divide a polynomial z3– 4z2 + 2z – 3 by z + 2. The steps required are as follows:
We first have to make sure that the polynomial is written in descending order. In our case, the question has already provided the expression in the required form. If there are any missing terms, we substitute them with zero.
We now have to divide the highest power of the dividend polynomial, that is, z3 in our case with the highest power of the divisor polynomial, i.e., z. When we do this, we get z2 that becomes the first term of the quotient.
Now we have to multiply this term with our divisor. z2 (z + 2) = z2 + 2z2
After multiplication, the resultant term has to be subtracted from the dividend.(z2– 4z2 + 2z – 3) – (z3 + 2z2) = -6z2 + 2z – 3.
The remainder term now becomes the dividend, and we repeat steps 2, 3, 4 until there are no more terms to bring down. This implies that we divide -6z2 by z to get -6z which forms term number two of our original quotient. Next, we multiply this with our divisor to get -6z (z + 2) = -6z2 – 12z. Now we subtract this from -6z2 + 2z – 3 to get 14z – 3. We have to perform the same steps one last time. On dividing 14z by z, we get 14 that forms the final term of our quotient. Multiplying this with the divisor and then subtracting the term, we get the remainder as 31.
Divisor = z + 2
Quotient = z2 – 6z + 14
Remainder = – 31
We now express this as z2 – 6z + 14 –
The technique of long division is used not only in polynomials but in other topics also; hence it is crucial to understand the concept well. By joining a platform such as Cuemath, kids can have the opportunity to avail themselves of excellent quality of education. The certified math tutors use several resources such as worksheets, workbooks, puzzles, etc., to teach kids and give them an enjoyable experience. Hopefully, this article helps you to strengthen your foundation in polynomials and long division!