Polynomial in Standard Form

The standard form of a polynomial is a polynomial written in the descending order of its exponents. 

Its general form with degree n is expressed as:

a_nxⁿ + a_{n – 1}x^{n – 1} + … + a₁x¹ + a₀ 

Here, 

• n is the highest degree
• a_nxⁿ is the leading term 

Thus, the term with the highest power is written first, followed by the terms with the second highest power, which continues and ends with the constant term.

Note: When written in standard form, the leading term is the first term of the polynomial, containing the highest degree of the exponent.

Here are a few examples of polynomials in standard form.

• 3x³ + 5x² – 8
• x⁴ – 5x + 6
• 5x⁷ – 2x⁵ + x² – 9x + 1

However, the following are NOT in standard form:

• 9 – x + x⁴ + 5x² 
• 8x – 3x² + 4x⁵ – 6
• -2x + 6 – x³ 

Degree

The degree of a polynomial represents the value of the largest exponent in the polynomial. When the polynomial is written in the standard form, the degree of the polynomial refers to the degree of the leading term (usually the first term). 

However, it differs based on the number of variables present in a polynomial. 

• ​For a polynomial with a single variable, the degree is the highest exponent of that variable.
• For a polynomial with multiple variables, the degree is the highest sum of the exponents of the variables in any single term.

For example,

5x⁷ – 2x⁵ + x² – 9x + 1 has a degree of 7 (as the highest exponent of the variable x is 7)

Now, let us consider a polynomial with 2 variables x³y² + x⁴ + 5x⁴y² – y² + 7

Here, 

Degree of x³y² = 3 + 2 = 5

Degree of x⁴ = 4

Degree of 5x⁴y² = 4 + 2 = 6

Degree of y² = 2

Degree of 7 = 0

Thus, the degree of x³y² + x⁴ + 5x⁴y² – y² + 7 = 6 (as the highest exponent among all terms is 6)

How To Write

Writing polynomials in standard form helps to identify the like terms in the given polynomials, which makes the simplification of polynomials much easier.

Let us consider the polynomial x³ + 4x⁴ – 3y² + 5xy – y² + 11x⁴ + 2x³ – 7 and convert it to standard form with the following steps.

Grouping Like Terms and Identifying the Number of Terms 

x³ + 4x⁴ – 3y² + 5xy – y² + 11x⁴ + 2x³ – 7

Combining all like terms, we get

= (x³ + 2x³) + (4x⁴ + 11x⁴) + (-3y² – y²) + 5xy – 7

= 3x³ + 15x⁴ – 4y² + 5xy – 7

Finding the Degree of Each Term

Here, 

The term 3x³ has a degree of 3

The term 15x⁴ has a degree of 4

The term -4y² has a degree of 2

The term 5xy has a degree of 2 (= 1 + 1)

The term -7 has a degree of 0

Arranging the Terms in Descending Order of the Degree

Thus, the polynomial in standard form is:

15x⁴ + 3x³ + 5xy – 4y² – 7

Types of Polynomials

Polynomials in standard form are classified into the following types based on the degree of their terms

Polynomial	Degree	Standard Form	Example
Constant	0	f(x) = c	8
Linear	1	f(x) = ax + b	8x – 3
Quadratic	2	f(x) = ax2 + bx + c	7x2 – 2x + 1
Cubic	3	f(x) = ax3 + bx2 + cx + d	x3 – 5x2 + 3x + 4
Quartic	4	f(x) = ax4 + bx3 + cx2 + dx + e	-x4 + x3 – 4x + 1
Quintic	5	f(x) = ax5 + bx4 + cx3 + dx2 + ex + f	3x5 + 2x3 – 5x + 4

Note: Polynomials with more than 5 degrees are called higher-degree polynomials.

Adding and Subtracting

When adding or subtracting polynomials, we add or subtract the like terms. 

Let us consider the polynomial expressions 2x³y² + x² and -5x² + x³y² 

Adding

(2x³y² + x²) + (-5x² + x³y²)

Writing the Polynomials in Standard Form

= (2x³y² + x²) + (x³y² – 5x²)

Combining the Like Terms Horizontally or Vertically and Then Adding

Thus, the sum is 3x³y² – 4x² 

Subtracting

(2x³y² + x²) – (-5x² + x³y²)

Writing the Polynomials in Standard Form

= (2x³y² + x²) – (x³y² – 5x²)

Combining the Like Terms Horizontally or Vertically and Then Subtracting

Thus, the difference is x³y² + 6x² 

Solved Examples