Polynomial Formula

Polynomial Formula specifies the standard form of polynomial expressions. It specifies the arrangement of algebraic expressions according to their increasing or decreasing power of variables. A polynomial is a combination (addition and/or subtraction) of repetitive monomial and/or binomial terms. The degree of a polynomial can be calculated by finding the highest power of the variable in the given polynomial. 

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

What is a Polynomial?

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

Let’s discuss below different types of polynomials in detail,

Types of Polynomial

There are four types of Polynomials which are,

• Monomial: Polynomials with one term are called monomials. Example: x, y², 3y³, etc.
• Binomial: Polynomials with two terms are called binomials. Example: 2x+y², x+3y³, etc.
• Trinomial: Polynomials with three terms are called trinomials. Example: 2x+z+y², z-x+3y³, etc.
• Quadratic Polynomial: Polynomials with four terms are called quadratic polynomials. Example: 2x+z³+y²+3y³, etc.

What is Polynomial Formula?

A polynomial is a combination of different algebraic expressions, The standard form of a polynomial consists of a combination of algebraic expressions and arranging them in ascending/descending order of power is called the Standard form of a polynomial. The standard form of a polynomial is called the polynomial formula. The polynomial formula is explained in detail below in this article,

Polynomial Formula

Polynomial Formula 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 various manners to get various algebraic expressions that are widely used. Some of the important formulas are,

f(x) = a_n(xⁿ)

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

Some other important polynomial formula is,

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

Polynomial Identities

Some of the polynomial identities which are widely used are discussed below,

• (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 + c² + 2xy + 2yz + 2zx

Applications of Polynomial Formula

Polynomial Formula has various applications some of its important applications are,

• Polynomials 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.
• Polynomials are used to estimate the curves of the roller-coaster tracks to estimate the suitable curvature and height of the tracks.
• Polynomials are used to correctly estimate the stock markets and accordingly shares can be purchased or sold.

Read More

• Vector Algebra
• Algebra Formulas
• Algebraic Identities of Polynomials

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:

FAQs on Polynomial Formula

Question 1: What is the Polynomial Formula in Algebra?

Answer:

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

Question 2: What is n In the Polynomial Formula?

Answer:

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

Question 3: What is the polynomial formula for Quadratic Polynomial?

Answer:

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.

Question 4: What is the cubed polynomial formula?

Answer:

The cubed polynomial formula is,

(x+y)³ = x³ + y³ + 3xy (x + y)

Question 5: What is the degree of the polynomial formula?

Answer:

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