A polynomial in standard form is written with the terms in order from highest degree, to lowest degree. For example, the polynomial 4x3 + 8x2 + 2x – 21 can be written in standard form as:
Generally, speaking, the standard form of a polynomial equation can be written as:
In other words, set your polynomial equal to 0 and write the terms in descending order by degrees. Sometimes (depending on your professor) it’s okay to omit the “= 0” part of the equation. Usually the clue is in the wording of the question:
- If you are giving an equation already set equal to zero, then include “= 0” in your answer.
- If you are given an equation that doesn’t include “= 0”, it’s usually okay to omit it.
For example, 5 + 9x3 -2x = 0 written in standard form is
9x3 -2x + 5 = 0.
But p(x) = 5 + 9x3 -2x in standard from is
p(x) = 9x3 -2x + 5.
Polynomial in Standard Form: A Few Rules
The variable x is always assigned a degree of 1 and a constant term is always assigned a degree of zero (the “degree” doesn’t exist for a constant as there is no variable). Degree 0 polynomials are sometimes called constant polynomials .
If you have more than one variable in a term, make sure to add all of the degrees up to get the correct degree. For example, 3x5 + 5x3y + 4y5z2 is written as a polynomial in standard form as:
4y5z2 + 3x5 + 5x3y.
Example question: Which of these polynomials is in standard form?
- 4 + 5x
- 5x + 3 – x2
- -21x5 + 3x3 – 99.
Solution: The third expression (-21x5 + 3x3 – 99) is in standard form because it is written in order of degree,
Example question #2: Write the following polynomial in standard form and classify the expression by degree: 3x2 – 7x
Step 1: Write the degree of each term:
- 3x2 = 2nd degree
- – 7x
4 = 4th degree
- + 38 = zero degree
- + 4x = 1st degree.
Step 2: Reorder the terms by order of degree:
 Wortman, K. Polynomials in Two Variables.