

What is a Polynomial Identity?
Sometimes, you come across polynomial problems that defy rationality. They just look at you, challenging you to think of a focus. There seems to be no intuitive way to solve the problem. In many of these situations, a polynomial identity has been created to slam the door open and ease the problem.
All identities of polynomials are essentially just true equations, but they are very helpful for demonstrating the connection between two seemingly unrelated expressions. Since the identity allows you to make that connection, you can use it to convert back and forth between the two expressions.
Some Useful Identities
There are many popular polynomial identities in the world of Mathematics, and here are some valuable ones:
This can speed up your FOIL (First – Outer – Inner – Last) factoring and multiplication. When a binomial is squared, it always breaks down into the same expression. A similar identity is one in which the terms are subtracted:
When you see a polynomial in any form on the right-hand side of the identity, you know that it will be included in the expression on the left. Remember, these are not the same as
The identity for cubes of the sum of two variables is
The identity for cubes of difference of two variables is
Difference Between Squares
The difference between the square’s identity can save you from factoring and multiplication:
This identity is so useful that you will find yourself looking through your polynomial problems, delighted when you see any form of difference between squares. Again, the sum of squares
Sum/Difference of Cubes
The lifesavers in terms of polynomial factorization are the sum of cubes and the difference of cube identities. These are extremely helpful to remember.
Quadratic formula
Although it's not as fun trying to remember, it is one of its most useful polynomial identities. Allows you to divide any quadratic expression (polynomial with a squared term as its highest order expression) into its factors:
If
Here are a couple of common polynomial identities that may be helpful while wandering through algebraic minefields, especially when working with quadratic equations:
(a + b) (c + d) = ac + ad + bc + bd (the FOIL formula)
Now let’s see some polynomial identity examples.
Solved Examples
Example 1: Solve the expression
Sol: Put a=x and b=2 in
Example 2: Solve the expression (6+4) (6-4).
Sol: Put a=6 and b=4 in polynomial identity
=36-16
=20
Example 3: Solve the expression
Sol: Quadratic formula is
Compare
Here, a=1, b=6, c=2
So,
Conclusion
True equations with broad applications in your algebra studies are important polynomial identities. Your Maths homework will be simpler the more comfortable you are with these, like:
, orThe difference between the identity of squares:
The sum of the identity of the cubes:
The identity difference of the cubes:
The FOIL formula: (a + b) (c + d) = ac + ad + bc + bd
Another FOIL formula:
The quadratic formula, which allows you to decompose any quadratic expression into its factors.
Practice Questions
Q 1: Solve the expression using polynomial identities
Ans: 100
Q 2: Solve the expression
Sol: 9984






FAQs on Polynomial Identities
1. What is a polynomial?
In Mathematics, a polynomial is an algebraic expression made up of numbers, letters, and exponents. That is, a polynomial consists of the addition or subtraction of different terms or monomials. The numbers in a polynomial are called coefficients, and the letters in a polynomial are its variables. The exponent of a variable is always a whole number and the coefficient of the variable is always a real number.
2. What are the parts of a polynomial?
Once we know what a polynomial consists of, we will see what they are and how to identify the parts of a polynomial:
Terms: Each monomial that is part of the polynomial.
Coefficients: The numbers that accompany each term of the polynomial.
Degree: The largest exponent to which the polynomial variable is raised.
Variable: Is the letter that the polynomial has.
Main term: It is the term with the highest degree of the polynomial.
Independent term: That term of the polynomial that does not have a variable.
Leading Coefficient: Coefficient of the leading term of the polynomial (highest power).
3. When is a polynomial said to be zero?
In Mathematics, a zero polynomial, also called a null polynomial, is a polynomial in which all its coefficients are equal to 0.





