Polynomial Identity

Polynomial Expressions That Agree for Every Input

An identity is an equation that remains true for every value in its domain. A polynomial identity is an identity whose two sides are polynomial expressions.

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

The following polynomial identities occur frequently:

Each identity can be read in both directions: left to right for expansion and right to left for factorization.

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):

   Find one value of the variable for which the two sides differ. A single counterexample proves that the equation is not an identity.

Examples of Proving Identities

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 $B = y^2$.

   $$\begin{aligned} &(2x^2-y^2)^2 \\ &=(2x^2)^2-2(2x^2)(y^2)+(y^2)^2 \end{aligned}$$

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

   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. Substitute one value for the variable, for instance $a = 0$, into both sides.

   • Left-Hand Side:

     $$\begin{aligned} &(2a - 5)(2a + 5) \\ &= (2(0) - 5)(2(0) + 5) \end{aligned}$$

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

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

     $$= -25$$

   • Right-Hand Side:

     $$\begin{aligned} &4a^2 - 20a + 25 \\ &= 4(0)^2 - 20(0) + 25 \end{aligned}$$

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

     $$= 0 - 0 + 25$$

     $$= 25$$

   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.

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

Exercise

Prove whether each of the following polynomial equations is a polynomial identity or not. A single value that breaks the equation is enough to show that it is not an identity, and expanding both sides is the way to show that one is.

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

Answer Key

1. 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 $B = 3$.

   Name the expanded square $B$, then retain the added constant:

   $$B=(2x)^2-2(2x)(3)+(3)^2$$

   $$(2x-3)^2+5=B+5$$

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

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

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

   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.