# Nakafa Framework: LLM

URL: /en/subject/high-school/11/mathematics/polynomial/polynomial-identity
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/polynomial/polynomial-identity/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Polynomial Identity",
  description: "Explore key polynomial identities like binomial squares and sum of cubes. Master these formulas to simplify expressions and solve equations efficiently.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/04/2025",
  subject: "Polynomial",
};

## 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, <InlineMath math="x + 2 = 5" /> is only true if <InlineMath math="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:

<div className="flex flex-col gap-4">
  <BlockMath math="a^2 - b^2 = (a + b)(a - b)" />
  <BlockMath math="(a + b)^2 = a^2 + 2ab + b^2" />
  <BlockMath math="(a - b)^2 = a^2 - 2ab + b^2" />
  <BlockMath math="a^3 + b^3 = (a + b)(a^2 - ab + b^2)" />
  <BlockMath math="a^3 - b^3 = (a - b)(a^2 + ab + b^2)" />
  <BlockMath math="(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3" />
  <BlockMath math="(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3" />
</div>

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. <InlineMath math="(2x^2 - y^2)^2 = 4x^4 - 4x^2y^2 + y^4" />
2. <InlineMath math="(2a - 5)(2a + 5) = 4a^2 - 20a + 25" />

**Solution:**

1. We will expand the left-hand side using the identity <InlineMath math="(A - B)^2 = A^2 - 2AB + B^2" />, with <InlineMath math="A = 2x^2" /> and <InlineMath math="B = y^2" />.

   <div className="flex flex-col gap-4">
     <BlockMath math="(2x^2 - y^2)^2 = (2x^2)^2 - 2(2x^2)(y^2) + (y^2)^2" />
     <BlockMath math="= 4x^4 - 4x^2y^2 + y^4" />
   </div>

   Since the result of expanding the left-hand side (<InlineMath math="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 <InlineMath math="a = 0" />, into both sides.

   - **Left-Hand Side:**

     <div className="flex flex-col gap-4">
       <BlockMath math="(2a - 5)(2a + 5) = (2(0) - 5)(2(0) + 5)" />
       <BlockMath math="= (0 - 5)(0 + 5)" />
       <BlockMath math="= (-5)(5)" />
       <BlockMath math="= -25" />
     </div>

   - **Right-Hand Side:**

     <div className="flex flex-col gap-4">
       <BlockMath math="4a^2 - 20a + 25 = 4(0)^2 - 20(0) + 25" />
       <BlockMath math="= 4(0) - 0 + 25" />
       <BlockMath math="= 0 - 0 + 25" />
       <BlockMath math="= 25" />
     </div>

   Since for <InlineMath math="a = 0" />, the left-hand side (<InlineMath math="-25" />) is not equal to the right-hand side (<InlineMath math="25" />), this equation is **not a polynomial identity**.

   Actually, the correct identity for <InlineMath math="(2a-5)(2a+5)" /> is <InlineMath math="(2a)^2 - 5^2 = 4a^2 - 25" />, using the identity <InlineMath math="(A-B)(A+B) = A^2-B^2" />.

## Exercise

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

1. <InlineMath math="(2m - 3)^3 = 8m^3 - 27" />
2. <InlineMath math="(2x - 3)^2 + 5 = 4x^2 - 12x + 14" />

### Answer Key

1. Let's test with the value <InlineMath math="m = 1" />.

   - **Left-Hand Side:**

     <div className="flex flex-col gap-4">
       <BlockMath math="(2m - 3)^3 = (2(1) - 3)^3" />
       <BlockMath math="= (2 - 3)^3" />
       <BlockMath math="= (-1)^3" />
       <BlockMath math="= -1" />
     </div>

   - **Right-Hand Side:**

     <div className="flex flex-col gap-4">
       <BlockMath math="8m^3 - 27 = 8(1)^3 - 27" />
       <BlockMath math="= 8(1) - 27" />
       <BlockMath math="= 8 - 27" />
       <BlockMath math="= -19" />
     </div>

   Since for <InlineMath math="m=1" />, the left-hand side (<InlineMath math="-1" />) <InlineMath math="\neq" /> the right-hand side (<InlineMath math="-19" />), this equation is **not a polynomial identity**.
   
   The correct identity is <InlineMath math="(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 <InlineMath math="(A - B)^2 = A^2 - 2AB + B^2" />, with <InlineMath math="A = 2x" /> and <InlineMath math="B = 3" />.

   <div className="flex flex-col gap-4">
     <BlockMath math="(2x - 3)^2 + 5 = [(2x)^2 - 2(2x)(3) + (3)^2] + 5" />
     <BlockMath math="= [4x^2 - 12x + 9] + 5" />
     <BlockMath math="= 4x^2 - 12x + 9 + 5" />
     <BlockMath math="= 4x^2 - 12x + 14" />
   </div>

   Since the result of expanding the left-hand side (<InlineMath math="4x^2 - 12x + 14" />) is exactly the same as the right-hand side, this equation is **proven to be a polynomial identity**.
