Polynomial Formula

Polynomial Formula gives the standard form of polynomial expressions. It specifies the arrangement of algebraic expressions according to their increasing or decreasing power of variables.

Let’s learn about various polynomial formulas and identities in detail.

What is Polynomial?

Algebraic expressions with non-negative power are called polynomials. Polynomials are the addition of monomials, binomials, and others.

The degree of a polynomial can be calculated by finding the highest power of the variable in the given polynomial. 

For example, f(x) = 3x² + 4x + 5 is a polynomial with a degree of 2. The polynomial formula has both like terms and unlike terms.

• Like terms have the same powers of the same variable.
• Unlike terms have different variables and different powers.

Types of Polynomial

Different Types of Polynomial have been discussed in the table below :

Type of Polynomial	Description	General Formula	Example
Monomial	Polynomials with one term	axn	x, y², 3y³, etc.
Binomial	Polynomials with two terms	axn+ bym	2x + y², x + 3y³, etc.
Trinomial	Polynomials with three terms	axn + bym + czk	2x + z + y², z – x + 3y³, etc.
Quadratic Polynomial	Second-degree polynomial (typically two or three terms)	ax2+ bx + c	2x + y² + 3y³, etc.

What is Polynomial Formula?

The standard form of a polynomia is called the polynomial formula. It consists of a combination of algebraic expressions arranging in ascending/descending order of powers.

Polynomial Formula

The General Formula of Polynomial is :

axⁿ + bx^n-1 + cx^n-2 +…+ px + q

where,
a, b, c, …, p and q are coefficients,
x is the variable,
n is the degree of the polynomial.

The above formula can be easily arranged in the following manner:

f(x) = a_n(xⁿ)

where
a is the coefficient,
x is the variable,
n is the exponent.

Some other important polynomial formula are :

• f(x) = a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + … + a₁x +a₀
• f(x) = a_nxⁿ + … + px + q

Quadratic Equation Formula

The standard form of Quadratic Equation is given by :

ax²+ bx + c

Cubic Equation Formula

The standard form of Cubic Equation is given by :

ax³ + bx²+ cx + d

Polynomial Identities

Let’s learn some of the polynomial identities and their expansion.

Identity	Expansion
(x + y)²	x² + 2xy + y²
(x – y)²	x² – 2xy + y²
x² – y²	(x + y)(x – y)
(x + y)³	x³ + y³ + 3xy(x + y)
(x – y)³	x³ – y³ – 3xy(x – y)
x³ + y³	(x + y)(x² – xy + y²)
x³ – y³	(x – y)(x² + xy + y²)
x³ + y³ + z³ – 3xyz	(x + y + z)(x² + y² + z² – xy – yz – zx)
(x + a)(x + b)	x² + (a + b)x + ab
(x + y + z)²	x² + y² + z² + 2xy + 2yz + 2zx

Applications of Polynomial Formula

Polynomial Formula has the following applications :

• They are used to define the equation of various forces, paths, and other concepts in detail.
• Polynomial equations are used to explain unknown quantities and their relation with other quantities in detail.
• Polynomial formulas are used to solve various complex mathematical equations.
• They are used to estimate the curves of the roller-coaster tracks to estimate the suitable curvature and height of the tracks.
• They are used to correctly estimate the stock markets and accordingly shares can be purchased or sold.

Related :

• Fundamentals of Polynomial
• Types of Polynomials
• Polynomial Function

• Algebraic Identities of Polynomials
• Algebra Formulas
• Algebraic Expressions
• Vector Algebra

Solved Examples on Polynomial Formula

Example 1: Find the factors of the given polynomial x² + 5x + 6

Solution:

Given polynomial,

x² + 5x + 6

= x² + 2x + 3x + 6

= x(x + 2) + 3(x + 2)

= (x + 2)(x + 3)

So factors of given polynomial are (x + 2) and (x + 3)

Example 2: Find the factors of the given polynomial x² + 3x – 4

Solution:

Given polynomial,

x² + 3x – 4

= x² + 4x – x – 4

= x(x + 4) – 1(x + 4)

= (x + 4)(x – 1)

So factors of given polynomial are (x + 4) and (x – 1)

Example 3: Find the factors of the given polynomial x² – 7x + 12

Solution:

Given polynomial,

x² – 7x + 12

x² – 4x – 3x + 12

x(x – 4) – 3(x – 4)

(x – 4)(x – 3)

So factors of given polynomial are (x – 4) and (x – 3)

Example 4: Simplify (x² + 6x + 9) / (x + 3)³

Solution:

Given, (x² + 6x + 9) / (x + 3)³

Now simplifying, 

x² + 6x + 9

= x² + 3x + 3x + 9

= x(x + 3) + 3(x + 3)

= (x + 3)(x + 3)

= (x + 3)²

 (x² + 6x + 9) / (x + 3)³ = (x + 3)² / (x + 3)³ 

                                    = 1/(x+3)

Example 5: Expand (3x – 11)³ using the cubic polynomial formula.

Solution:

We know that, (x – y)³= x³ – y³ – 3xy (x – y)

Now, (3x – 11)³

= (3x)³ – (11)³ – 3(3x)(11)(3x-11)

= 27x³ – 1331 – 9x(3x -11)

This is the required expansion.

Example 6: Divide the polynomial x³ – 6x² +3x + 10 by x + 1

Solution:

Polynomial Formulas- FAQs

What is the Polynomial Formula in Algebra?

Polynomial Formula consists of the different combinations of algebric expressions arranged in the ascending/descending forms. A standard form of the polynomial formula can be given as, 

axⁿ + bx^n-1 + cx^n-2 +…..+ px + q

What is n in Polynomial Formula?

For any polynomial, the power of the highest term is called as the degree of the polynomial. It is denoted as n.

What is the formula of Quadratic Polynomial?

A polynomial with degree 2 is called a quadratic polynomial. The polynomial formula for Quadratic Polynomials is,

ax² + bx + c 

where 
a, b and c are the real numbers.

What is the polynomial formula cubed ?

The cubed polynomial formula is,

(x+y)³ = x³ + y³ + 3xy (x + y)

What is the degree of polynomial?

The term with the highest power in any polynomial formula is called the degree of the polynomial.