POLYNOMIALS IN STANDARD FORM

The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form.

The standard form of a polynomial that contains one variable is written with the terms in order from greatest degree to least degree 

Leading Coefficient of a Polynomial

The leading coefficient of a polynomial is the coefficient of the term with the highest power of the variable. 

For example, the leading coefficient of 5 - x - + 5x² - 3x⁴ is -3. 

When a polynomial is written in standard from, the coefficient of the first term is called the leading coefficient. 

Let us consider the polynomial given below. 

7y³ + 2y² - 3y - 1

The above polynomial is in standard form. 

The leading coefficient is the coefficient of the first term, 7.

Writing Polynomials in Standard Form

Write each polynomial in standard form. Then, give the leading coefficient.   

Example 1 :

20y + 4y³ - 2 + y²

Solution : 

Find the degree of each term :

20y  :  degree 1

4y³  :  degree 3

-2  :  degree 0

y²  :  degree 2

Arrange the terms in order from largest degree to smallest degree.

4y³ + y² + 20y - 2

The standard form is (4y³ + y² + 20y - 2) and the leading coefficient is 4.

Example 2 :

x³ + x⁵ + 4x

Solution : 

Find the degree of each term :

x³  :  degree 3

x⁵  :  degree 5

4x  :  degree 1

Arrange the terms in order from largest degree to smallest degree.

x⁵ + x³ + 4x

The standard form is (x⁵ + x³ + 4x) and the leading coefficient is 1.

Example 3 :

5x - 17x⁶

Solution : 

Find the degree of each term :

5x  :  degree 1

-17x⁶  :  degree 6

Arrange the terms in order from largest degree to smallest degree.

-17x⁶ + 5x

The standard form is (-17x⁶ + 5x) and the leading coefficient is -17.

Example 4 :

-6x⁴ - 5x - 9x³

Solution : 

Find the degree of each term :

-6x⁴  :  degree 4

-5x  :  degree 1

-9x³  :  degree 3

Arrange the terms in order from largest degree to smallest degree.

-6x⁴ - 9x³ - 5x

The standard form is (-6x⁴ - 9x³ - 5x) and the leading coefficient is -6.

Example 5 :

9x³ - x + 3 - 7x⁴

Solution : 

Find the degree of each term :

9x³  :  degree 3

-x  :  degree 1

3  :  degree 0

-7x⁴  :  degree 4

Arrange the terms in order from largest degree to smallest degree.

-7x⁴ + 9x³ - x + 3

The standard form is (-7x⁴ + 9x³ - x + 3) and the leading coefficient is -7.

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