How to Write Polynomials in Standard Form

How to Write Polynomials in Standard Form

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What is a Polynomial?

A polynomial can be broken down into two words: "poly" and "nomial," which both signify many  terms, as suggested by the name of the concept. In mathematics, a polynomial is an algebraic or mathematical expression that has at least two terms that are linked together by a mathematical operation. It should be noted that expressions within the polynomial terms could be a constant, a variable, or even a mixed term that contains a co-efficient. A polynomial should also have the power of the variable (let's say $x$) be an integer rather than a fractional power such as the squared root or the cube root.
For example, $7x^{\frac{1}{2}} \ - \ 5x \ + \ 13$ is not a polynomial as the first term has a fraction power of $x$.

Different Types of Polynomials

Based on the number of terms, there are 3 different types of polynomials. They are monomials, binomials and trinomials. Let’s discuss more about them:

• Monomial: They consist of just a single term but there is one condition. The term cannot be zero. Also, monomials can all be mixed terms as well as just pure variable terms. Ex: $5x^2, \ 6x^3, \ 69x$, etc.
• Binomial: Binomials consist of just two terms and they must have a mathematical operation in between. Binomials can have mixed terms, pure variable terms, or even pure constant terms in the expression. Ex: $2x^2 \ + \ 5x^3, \ 3x^2 \ + \ 8y, \ xy^2 \ + \ 2x$, etc.
• Trinomial: Trinomial consist of three terms and between every two terms there is a mathematical operation. Trinomials too can have mixed terms, pure variable terms, and even pure constant terms in the expression. Ex: $2x^2 \ + \ 3x \ + \ 10, \ 4x^3 \ + \ 3x^2 \ + \ 7, \ 3x^2y \ + \ 4xy \ + \ 6$, etc.

Standard Form of Polynomials

The standard form of a polynomial function $f(x)$ of the degree $n$ is:
$F(x) \ = \ a_{n}x^n \ + \ a_{n-1}x^{n \ - \ 1} \ + \ …… \ + \ a_{1}x \ + \ a_{0}$, where the first term is the term with the biggest power of the variable (in this case $x$).
To write a polynomial in the standard form, we must arrange all powers of $x$ in descending order. For example, in $4x^2 \ - \ 9x^3 \ + \ 13x \ - \ 7$, we write the standard form as:
$-9x^3 \ + \ 4x^2 \ + \ 13x \ – \ 7$.

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