Explore: Polynomial Identities Solutions

An identity is an equation that will be true for any value of variable(s) in the equation. 
This means that the solution will be true for any value that replaces the variable. 
Recognizing a polynomial identity can help simplify one or both sides of an equation. 
This is because when the identity exists, the sides of the equation are interchangeable because they are equal. 
Some polynomial identities that you are already familiar with are:
- - Difference of two squares: $a^{2} - b^{2} = (a - b) (a + b)$
  - Perfect square trinomials:
    ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$
    ${(a - b)}^{2} = a^{2} - 2 a b + b^{2}$
  - Sum of cubes: $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$
  - Difference of cubes: $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$

Note

“Value” is used rather than all real numbers because you will learn in subsequent units that real, complex numbers and non-real, complex numbers are true for polynomial identities.

Example 1

Determine if a polynomial identity exists. Explain.
${(x^{2} + y^{2})}^{2} = {(x^{2} - y^{2})}^{2} + {(2 x y)}^{2}$

Implement

Left side	Right side
${(x^{2} + y^{2})}^{2}$	${(x^{2} - y^{2})}^{2} + {(2 x y)}^{2}$
$(x^{2} + y^{2}) (x^{2} + y^{2})$	$(x^{2} - y^{2}) (x^{2} - y^{2}) + 2^{2} x^{2} y^{2}$
$x^{4} + x^{2} y^{2} + x^{2} y^{2} + y^{4}$	$x^{4} - x^{2} y^{2} - x^{2} y^{2} + y^{4} + 4 x^{2} y^{2}$
$x^{4} + 2 x^{2} y^{2} + y^{4}$	$x^{4} + 2 x^{2} y^{2} + y^{4}$

Explain

Write each side of the problem 
Expand expressions 
Distribute 
Combine like terms

When written in the same form, the left and right sides of the equation have identical terms, which means this represents a polynomial identity .

Note

This polynomial identity can be used to generate Pythogorean triples for right triangles.

Example 2

Determine if a polynomial identity exists. Explain.
${(x - 5)}^{2} = (x - 4) (x + 4) + (x - 3) (x + 3)$

Left side	Right side
${(x - 5)}^{2}$	$(x - 4) (x + 4) + (x - 3) (x + 3)$
$(x - 5) (x - 5)$	$x^{2} + 4 x - 4 x - 16 + x^{2} + 3 x - 3 x - 9$
$x^{2} - 5 x - 5 x + 25$	$2 x^{2} - 25$
$x^{2} - 10 x + 25$

The sides of the equation are not equal; therefore, this does not represent an identity.

Example 3

Determine if a polynomial identity exists. Explain.
${(a - b)}^{3} = a^{3} - b^{3} - 3 a b (a - b)$

Left side	Right side
${(a - b)}^{3}$	$a^{3} - b^{3} - 3 a b (a - b)$
$(a - b) (a - b) (a - b)$	$a^{3} - b^{3} - 3 a^{2} b + 3 a b^{2}$
$(a^{2} - 2 a b + b^{2}) (a - b) \begin{matrix} a^{3} & - 2 a^{2} b & + a b^{2} \end{matrix} + \begin{matrix} - a^{2} b & + 2 a b^{2} & - b^{3} \end{matrix} \begin{matrix} a^{3} & - b^{3} & - 3 a^{2} b & + 3 a b^{2} \end{matrix}$

Since the left and right side of the equation have identical terms when written in the same form, this represents a polynomial identity.

Note

You should write the left and right side expressions in the same form to help determine if an identity exists. However, identities do not have to be in standard form, or even in the same order IF you can explain:

that the terms can be arranged in any order using the commutative property, and
result in the sides being equal.