Polynomial in standard form

On this post you will find the explanation of what a polynomial in standard form is. You will also see examples of polynomials in standard form and how to put a polynomial in standard form. And finally, you will find solved practice problems on writing a polynomial in standard form.

When is a polynomial in standard form?

A polynomial is in standard form when all its terms are arranged by decreasing order of degree, that is, a polynomial in standard form is a polynomial whose terms are all ordered from highest to lowest degree.

For example, the following polynomial is in standard form:

x^4+5x^3-4x^2+3x+6

As you can see, the previous polynomial is in standard form because its terms are arranged by decreasing order. In other words, first we have the term x4 which is of fourth degree, secondly there is 5x3 which is of third degree, then -4x2 which is of second degree, then 3x which is of first degree and finally 6 which is the constant term of the polynomial (degree equal to 0).

Therefore, every polynomial whose terms are not ordered in descending order it is not in standard form:

6x^2+5x+2x^6+4-9x^5  ❌

Note: there are math books that consider that a polynomial is in standard form when its terms are written in increasing order, such as the following polynomial:

2-x+6x^2+7x^3

However, the most common is to refer to a polynomial in standard form when its terms are in decreasing order.

Although the concept of a polynomial in standard form seems to be a very simple, you should know that this is very important to perform some polynomial operations. For example, the result of a polynomial division will be wrong if the polynomials are not placed in standard form.

➤ See: how to do a long division of polynomials

Examples of polynomials in standard form

Once we have seen the definition of a polynomial in standard form, let’s see several examples of polynomials in standard form to understand better its meaning.

  • Example of a polynomial in standard form without constant term:

x^5+2x^3 + 6x

As you can see in the previous example, it is not necessary for a polynomial in standard form to have all the terms of all degrees, as long as all its terms are ordered from highest to lowest degree it will be considered a polynomial in standard form. Thus, the above example does not have a term of degree 4, nor a term of degree 2, nor a constant term, but it is in standard form because all its monomials are ordered in descending order.

  • Example of a monic polynomial in standard form:

x^4+3x^3-5x+7

  • Example of a polynomial in standard form with terms of all degrees:

3x^6+x^5-6x^4+x^3+2x^2-9x+1

How to write a polynomial in standard form

To write a polynomial in standard form, you must do the following steps:

  1. Add (or subtract) the like terms of the polynomial.
  2. Write the term with the highest degree first.
  3. Write all the other terms in decreasing order of degree.
  4. Remember that a term with a variable but without an exponent is of degree 1.
  5. Remember that a constant term is of degree 0, so it always is the last term.

We are going to rewrite the following polynomial in standard form as an example:

1-4x^3+x^4+2x

In this case the polynomial has no like terms, so we don’t have to do any addition or subtraction.

The highest exponent is 4, thus the first term of the polynomial must be x4.

The next highest degree term is -4x3, so we put it next: x4-4x3.

There is no second degree term, therefore the next element is 2x, which is of first degree. Then we have x4-4x3+2x.

And the constant term goes last. So the polynomial expressed in standard form is x4-4x3+2x+1.

x^4-4x^3+2x+1

Practice problems on writing polynomials in standard form

Problem 1

Write the following polynomial in standard form:

2x^3-5+3x^2+4x^5-3x^6+7x

There are no like terms in the polynomial, so we must not add or subtract any terms. Therefore, to put the polynomial in standard form we simply have to arrange its terms in decreasing order of exponent:

-3x^6+4x^5+2x^3+3x^2+7x-5

 

Problem 2

Put the following polynomial in standard form:

3x^2+2+2x-5x^3+5x^2-7x+1

First of all, we have to add and subtract the like terms of the polynomial:

(3x^2+5x^2)+(2-1)+(2x-7x)-5x^3

8x^2+1-5x-5x^3

And once we have grouped the terms of the same degree, we put them in descending order:

-5x^3+8x^2-5x+1

 

Problem 3

Rewrite the following polynomial with two variables in standard form:

1+4x^2y-y-8xy-x^4y^3+5x^4-y^{10}+x^4y+7x^3y^5

To find the degree of a term with two or more variables, you have to add the exponents of all the variables of the term. For example, the term 7x3y5 is of degree 8, since 3+5=8.

Thus, the terms arranged in decreasing order of degree from left to right (polynomial in standard form) are as follows:

-y^{10}+7x^3y^5-x^4y^3+x^4y+5x^4+4x^2y-8xy-y+1

 

Leave a Comment

Your email address will not be published.

Scroll to Top