Polynomial Identity

What is a Polynomial Identity?

Have you ever seen a mathematical equation that is always true, no matter what value we substitute for the variable? That's called an identity. Well, a Polynomial Identity is an identity that involves polynomial forms.

Unlike a regular equation which is only true for specific variable values (for example, $x + 2 = 5$ is only true if $x=3$), a polynomial identity holds true for all possible values of the variable.

Commonly Used Polynomial Identities

Here are some important and frequently encountered polynomial identities:

These identities are very useful for simplifying or factoring polynomial expressions.

Proving an Equation is an Identity

How do we know if an equation is truly an identity or not?

1. How to Prove (If it IS an identity):

   We must show that the expression on the left-hand side of the equation is always equal to the expression on the right-hand side after simplification. This is done by expanding one side (usually the more complex one) using algebraic operations until its form exactly matches the other side.

2. How to Disprove (If it is NOT an identity):

   Simply find one example value for the variable that makes the left-hand side not equal to the right-hand side. If we can find just one value that makes the equation false, then it is not an identity.

Proving Identities: Examples

Prove whether the following equations are polynomial identities or not.

1. $(2x^2 - y^2)^2 = 4x^4 - 4x^2y^2 + y^4$
2. $(2a - 5)(2a + 5) = 4a^2 - 20a + 25$

Solution:

1. We will expand the left-hand side using the identity $(A - B)^2 = A^2 - 2AB + B^2$, with $A = 2x^2$ and $B = y^2$.

   $$(2x^2 - y^2)^2 = (2x^2)^2 - 2(2x^2)(y^2) + (y^2)^2$$

   $$= 4x^4 - 4x^2y^2 + y^4$$

   Since the result of expanding the left-hand side ($4x^4 - 4x^2y^2 + y^4$) is exactly the same as the right-hand side, this equation is proven to be a polynomial identity.

2. Let's substitute one value for the variable, for instance $a = 0$, into both sides.

   • Left-Hand Side:

     $$(2a - 5)(2a + 5) = (2(0) - 5)(2(0) + 5)$$

     $$= (0 - 5)(0 + 5)$$

     $$= (-5)(5)$$

     $$= -25$$

   • Right-Hand Side:

     $$4a^2 - 20a + 25 = 4(0)^2 - 20(0) + 25$$

     $$= 4(0) - 0 + 25$$

     $$= 0 - 0 + 25$$

     $$= 25$$

   Since for $a = 0$, the left-hand side ($-25$) is not equal to the right-hand side ($25$), this equation is not a polynomial identity.

   Actually, the correct identity for $(2a-5)(2a+5)$ is $(2a)^2 - 5^2 = 4a^2 - 25$, using the identity $(A-B)(A+B) = A^2-B^2$.

Exercise

Prove whether each of the following polynomial equations is a polynomial identity or not.

1. $(2m - 3)^3 = 8m^3 - 27$
2. $(2x - 3)^2 + 5 = 4x^2 - 12x + 14$

Answer Key

1. Let's test with the value $m = 1$.

   • Left-Hand Side:

     $$(2m - 3)^3 = (2(1) - 3)^3$$

     $$= (2 - 3)^3$$

     $$= (-1)^3$$

     $$= -1$$

   • Right-Hand Side:

     $$8m^3 - 27 = 8(1)^3 - 27$$

     $$= 8(1) - 27$$

     $$= 8 - 27$$

     $$= -19$$

   Since for $m=1$, the left-hand side ($-1$) $\neq$ the right-hand side ($-19$), this equation is not a polynomial identity.

   The correct identity is $(2m-3)^3 = (2m)^3 - 3(2m)^2(3) + 3(2m)(3)^2 - 3^3 = 8m^3 - 36m^2 + 54m - 27$.

2. We will expand the left-hand side using the identity $(A - B)^2 = A^2 - 2AB + B^2$, with $A = 2x$ and $B = 3$.

   $$(2x - 3)^2 + 5 = [(2x)^2 - 2(2x)(3) + (3)^2] + 5$$

   $$= [4x^2 - 12x + 9] + 5$$

   $$= 4x^2 - 12x + 9 + 5$$

   $$= 4x^2 - 12x + 14$$

   Since the result of expanding the left-hand side ($4x^2 - 12x + 14$) is exactly the same as the right-hand side, this equation is proven to be a polynomial identity.