Polynomial Identity

Polynomial identities are algebraic expressions involving polynomials that are true for all values of the variables. These identities are used to simplify complex expressions and solve equations efficiently. A polynomial can include one or more variables, each raised to a positive integer power. Polynomial identities are equations that remain valid for all values of the variables involved.

Polynomial identities can be classified into different types based on the degree of the polynomials. The main types of polynomial identities are:

• Second Degree Polynomial Identities 

• Third Degree Polynomial Identities 

• n-Degree Polynomial Identities 

The polynomial identities have the highest degree of two. The second-degree polynomial identities are:

• (a + b)² = a² + 2ab + b²
   
• (a - b)² = a² - 2ab + b²
   
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
   
• (a - b - c)² = a² - b² - c2 - 2ab + 2bc - 2ca
   
• (x + a)(x + b) = x² + (a + b)x + ab

The third-degree polynomial identities include the polynomials with the highest degree of 3. Some examples of third-degree polynomial identities are:

• (a + b)³ = a³ + b³ + 3ab(a + b)
   
• (a - b)³ = a³ - b³ - 3ab(a - b)
   
• a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca) 

The polynomial with the highest power of the variable n is the n-degree polynomial. The formula for all n-degree polynomial identities is:

If n is a natural number: 

aⁿ - bⁿ = (a - b)[(a^{n - 1}) + (a^{n - 2})b + … + (b^{n - 2})a + (b^{n - 1})]

If n is an even number (n = 2k):

aⁿ + bⁿ= (a + b)[(a^{n - 1}) - (a^{n -2} )b + …. + (b^{n - 2})a - (b^{n - 1})]

If n is an odd number (n = 2k + 1)

aⁿ + bⁿ = (a + b)((a^{n - 1}) - (a^{n - 1}) - (a^{n - 2})b + …. - (b^{n - 2})a + (b^{n - 1}))

The polynomial identities are used to solve, simplify, and expand the polynomial expressions. Here are some formulas for polynomials:

• a² - b² = (a + b)(a - b)

• a³ - b³ = (a - b)(a² + ab + b²)

• a³ + b³ = (a + b)(a² - ab + b²)

• a⁴ - b⁴ = (a+ b)(a - b)[(a + b)² - 2ab]

• 2(a² + b²) = (a + b)² + (a - b)²

• (a + b)² - (a - b)² = 4ab

Now, let’s learn how to prove some commonly used polynomial identities. By learning these, students can easily understand how these identities work. In this section, we will learn the proofs for the given identities:

• (a + b)² = a² + 2ab + b² 
   
• (a - b)² = a² - 2ab + b²
   
• (a + b)(a - b) = a² - b²
   
• (x + a)(x + b) = x² + x(a + b) + ab

Identity 1: (a + b)² = a² + 2ab + b² 

Expanding the expression using distributive property: (a + b)² = (a + b) × (a + b)
= a² + ab + ba + b² 

ab = ba, since multiplication is commutative 
= a² + 2ab + b²

Therefore, (a + b)² = a² + 2ab + b² 

Identity 2: (a - b)² = a² - 2ab + b²

Using distributive property to prove (a - b)² = a² - 2ab + b². 

So, (a - b)² = (a - b)(a - b)

= a² - ab - ba + b² 

= a² - 2ab +b², as ab = ba

Thus, (a - b)² = a² - 2ab + b²

Identity 3: (a + b)(a - b) = a² - b²

To multiply two expressions, we use the distributive property. 

That is a(a - b) + b(a - b) = a² - ab + ab - b²
= a² - b²

So, (a + b)(a - b) = a2 - b2

Identity 4: (x + a)(x + b) = x² + x(a + b) + ab

To prove (x + a)(x + b) = x² + x(a + b) + ab

We expand, (x + a)(x + b) using the distributive property:
(x + a)(x + b) = x² + xa + xb + ab

Factoring out the common term (x) from xa + xb
So, x² + xa + xb + ab becomes:

x² + x(a + b) + ab

Hence, proved (x + a)(x + b) = x² + x(a + b) + ab

Students usually find problems involving polynomial identities difficult, but these identities are used in various ways to simplify and solve polynomial expressions. Here are a few tips to solve polynomial equations. 

Tip 1: When solving a problem, first check what information is given and what the question is asking for. This helps students identify the identity to be used.

Tip 2: Next, identify the degree of the equation and check for the related identities. For example, if the degree of the equation is 3, check for the identities related to the third-degree polynomial. 

Tip 3: After identifying the identity, substitute the values of the variable in the identity and solve the expression. 

Polynomial identities are used in different fields like computer science, physics, engineering, etc. Here are some applications of the polynomial identity 

• In computer graphics and animation, the polynomial identities are used to model curves, shapes, and animations. 

• In finance, to calculate compound interest and investment growth, we use polynomial identities. For example, the identity (1 + r)2 is used to calculate compound interest. 

• In construction, to estimate the cost or area when adding extra units to the base plan, we use polynomial identities. 

• In mathematics, to factor equations, find the roots of quadratic equations, simplify expressions, and solve systems of equations, we use polynomial identities.