Polynomial Formula

The 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.

The General Formula of a Polynomial:

f(x) = a_n​xⁿ + a_n−1​xⁿ⁻¹ + ⋯ + a₁​x + a₀​

Where,

• a_n​, a_n−1​, …, a₁​, a₀​ are the coefficients,
• x is the variable,
• n is the degree of the polynomial (the highest power of x).

What is Polynomial?

A polynomial is an algebraic expression consisting of terms with non-negative integer exponents of the variable. It can be expressed as the sum of monomials, binomials, or more complex expressions.

The degree of a polynomial is determined by the highest power of the variable in the expression. For example, in the polynomial f(x) = 3x² + 4x + 5, the highest power of x is 2, so the degree of the polynomial is 2.

Like and Unlike Terms:

• Like terms: Terms that have the same variable raised to the same power (e.g., 3x² and 5x²).
• Unlike terms: Terms that have different variables or different powers (e.g., 3x² and 4x).

Types of Polynomial

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

Read More: Types of Polynomials (Based on Terms and Degrees)

Polynomial Identities

Let's learn some of the algebraic identities of polynomials and their expansion.

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.

Read More: Real-Life Applications of Polynomials

Related :

• Polynomials
• Types of Polynomials
• Polynomial Function
• Algebra Formulas
• Algebraic Expressions

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)
= 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:

Check: Practice Questions on PolynomialsPolynomials

Practice Problems on Polynomials

Problem 1: Evaluate the polynomial at given values: P(x) = 3x⁴ − 5x³ + 2x² − x + 7

• P(1)
• P(−2)
• P(0)

Problem 2: Factor the polynomial completely: x³ − 6x² + 11x − 6

Problem 3: Find the roots of the polynomial: 2x² − 4x − 6 = 0

Problem 4: Simplify the polynomial expression: (3x² + 2x − 5) + (2x² − 3x + 4)

Problem 5: Multiply the polynomials: (2x − 3)(x² + x + 4)