Calculus How To

Polynomial in Standard Form

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Standard Form >

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:
polynomial in standard form


Generally, speaking, the standard form of a polynomial equation can be written as:

cn·xn + cn – 1·x – 1n + … + c1·x + c0 = 0

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 [1].

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.
degree of a polynomial

Example question: Which of these polynomials is in standard form?

  1. 4 + 5x
  2. 5x + 3 – x2
  3. -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 – 7x4 + 38 + 4x.

Solution:

Step 1: Write the degree of each term:

  • 3x2 = 2nd degree
  • – 7x4 = 4th degree
  • + 38 = zero degree
  • + 4x = 1st degree.

Step 2: Reorder the terms by order of degree:

– 7x4 + 3x2 + 4x + 38.

References

[1] Wortman, K. Polynomials in Two Variables.

CITE THIS AS:
Stephanie Glen. "Polynomial in Standard Form" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/polynomial-in-standard-form/
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