3. Polynomials

Polynomials

The product of a coefficient and a variable is called a term.

When two or more terms are added together, the result is a polynomial.

A polynomial with only one term is called a monomial. A polynomial with two terms is known as a binomial, whereas a polynomial with three terms is known as a trinomial. The degree of a polynomial is equal the degree of the term with the highest exponent.

Polynomial terms are usually arranged in such a way that the exponents of the variables decrease from left to right. A term without a variable is considered a constant, and it is written last. If a polynomial contains several different variables, they are presented in alphabetical order.

Example 1

a) The polynomial [[$ 2x^3 - x + 5 $]] is a trinomial with a degree of [[$ 3 $]].

b) The polynomial [[$ -2y $]] is a monomial with a degree of [[$ 1 $]].

c) The polynomial [[$ 4x ^ 2y + x $]] is a binomial with a degree of [[$ 2 $]].

Polynomial terms are said to have the same form if they contain the same variable. Only terms of the same form can be combined during addition and subtraction calculations.

The addition and subtraction of polynomials

In addition calculations, polynomials are written with their signs in sequence, and terms of the same form are combined with each other.
In subtraction calculations, a polynomial can be subtracted from another polynomial so that each term is subtracted separately. If there is a minus sign in front of the brackets, the signs of all terms must be reversed when the brackets are removed.

Two polynomials that have a sum of zero are said to be each others' opposite polynomials. The opposite polynomial is obtained by changing the sign of each term in the polynomial.

Example 2

Calculate the difference between the polynomials [[$ 2x ^ 2 - 3x + 3 $]] and [[$ -5x + 1 $]].

In the multiplication of polynomials, terms are multiplied together with their signs. It is a good idea to mark the intermediate steps carefully to avoid errors.

The multiplication of polynomials

When multiplying monomials, the coefficients are multiplied and the parts of the variables are multiplied.
A polynomial is multiplied by a monomial by multiplying each term of the polynomial is by the monomial separately. The resulting products are added together: [[$ a(b + c) = ab + ac $]]
In the multiplication of polynomials, each multiplier term multiplies each multiplicand term. The resulting products added together: [[$ (a + b)(c + d) = ac + ad + dc + bd $]]

Example 3

Calculate the product of the monomials [[$ 3x ^ 2 $]] and [[$ 4x ^ 4 $]].

Example 4

Calculate the product of the monomial [[$ 3x $]] and binomial [[$ 2x - 1 $]].

Example 5

Calculate the product of the polynomials [[$ 2x - 1 $]] and [[$ -4x + 1 $]].