# Polynomial identity: Explained in a nutshell

The polynomial identity is a mathematical fact or equation that helps us solve expressions with large numbers and exponents quickly. It simplifies an expression by breaking the numbers into simpler units. (a + b)(a − b) = a^{2} − b^{2} is a polynomial equation that contrasts with **polynomial identities**, where both expressions represent the same polynomial.

Therefore, any evaluation of both members is valid. (a + b)(a − b) = a^{2} − b^{2} is a polynomial equation, in contrast to a polynomial identity, where both expressions represent the same polynomial. Therefore, any evaluation of both members gives valid equality. In a polynomial, there is another concept called **zero polynomial**.

## What is a Polynomial Identity?

An equation with polynomial identity is always true regardless of the variables’ values. In order to expand or factorize polynomials, we use polynomial identities.

Polynomial Identity Examples

Consider these equations: 4x – 2 = 14 and 8x – 4 = 28. You will observe that x = 4 in both equations if you solve them separately. The two equations are ax – b = c if you write them as:

ax – b = c

2ax – 2b = 2c

Essential Polynomial Identities

In mathematics, polynomial identities are also known as algebraic identities.

(a+b)^{2} = a^{2}+b^{2}+2ab

(a-b)^{2} = a^{2}+b^{2}-2ab

(a+b)(a-b) = a^{2}-b^{2}

(x+a)(x+b) = x^{2}+ x(a+b)+ab

In addition to the above-mentioned polynomial identities, there are several other polynomial identities that are equally important when solving expressions. They are:

(a+b)^{3} = a^{3}+3a^{2}b+3ab^{2}+b^{3}

(a-b)^{3} = a^{3}-3a^{2}b +3ab^{2}– b^{3}

a^{3}+b^{3} = (a+b)(a^{2}-ab+b^{2})

a^{3}-b^{3} = (a-b)(a^{2}+ab+b^{2})

(a+b+c)^{2} = a^{2}+b^{2}+c^{2}+2ab+2bc+2ca

Use Cases of Polynomials

Polynomials in the Supermarket

In your head, you’ve probably used polynomials more than once while shopping. Three pounds of flour, two dozen eggs, and three quarts of milk, for example, might cost you $30. In order to check the prices, construct a simple polynomial, where “f” represents the price of flour, “e” represents the price of a dozen eggs, and “m” represents the price of a quart of milk. The formula is as follows: 3f + 2e + 3m.

Prices can now be input into this basic algebraic expression. At checkout, you will be charged 3(4.49) + 2(3.59) + 3(1.79) = $26.02, plus tax, for flour, eggs, and milk.

People Who Use Polynomials

Career professionals who need to perform complex calculations are most likely to use polynomials on a daily basis. In the design of roller coasters, polynomials are used to model curves, while in the design of buildings and roads, polynomials are used. Polynomials can also be used to describe and predict traffic patterns, so that appropriate traffic control measures, such as traffic lights, can be implemented. Polynomials are used by economists to model economic growth patterns, as well as by medical researchers to describe the behavior of bacterial colonies.

The use of polynomials can benefit even a taxi driver. If a driver wants to earn $100, he needs to drive how many miles. It can be expressed in polynomial form as 1/2 ($1.50)x if the meter charges the customer $1.50 a mile and the driver receives half of it. The answer to this polynomial is 133.33 miles if you allow it to equal $100 and solve for x.

So you can see that these polynomial identities are not limited to theoretical bounds but also have practical examples too. Follow this guide and understand what crucial identities are there and boost your knowledge.

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