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Explore key polynomial identities like binomial squares and sum of cubes. Learn these formulas to simplify expressions and solve equations efficiently.
---
## 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.
Visible text: Unlike a regular equation which is only true for specific variable values (for example, is only true if ), 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:
Component: MathContainer
Children:
```math
a^2 - b^2 = (a + b)(a - b)
```
```math
(a + b)^2 = a^2 + 2ab + b^2
```
```math
(a - b)^2 = a^2 - 2ab + b^2
```
```math
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
```
```math
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
```
```math
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
```
```math
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
```
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.
Visible text: 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. $$(2a - 5)(2a + 5) = 4a^2 - 20a + 25$$
Visible text: 1.
2.
**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 $$B = y^2$$.
```math
(2x^2 - y^2)^2 = (2x^2)^2 - 2(2x^2)(y^2) + (y^2)^2
```
```math
= 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:**
```math
(2a - 5)(2a + 5) = (2(0) - 5)(2(0) + 5)
```
```math
= (0 - 5)(0 + 5)
```
```math
= (-5)(5)
```
```math
= -25
```
- **Right-Hand Side:**
```math
4a^2 - 20a + 25 = 4(0)^2 - 20(0) + 25
```
```math
= 4(0) - 0 + 25
```
```math
= 0 - 0 + 25
```
```math
= 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 $$(A-B)(A+B) = A^2-B^2$$.
Visible text: 1. We will expand the left-hand side using the identity , with and .
Since the result of expanding the left-hand side () 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 , into both sides.
- **Left-Hand Side:**
- **Right-Hand Side:**
Since for , the left-hand side () is not equal to the right-hand side (), this equation is **not a polynomial identity**.
Actually, the correct identity for is , using the identity .
## Exercise
Prove whether each of the following polynomial equations is a polynomial identity or not.
1. $$(2m - 3)^3 = 8m^3 - 27$$
2. $$(2x - 3)^2 + 5 = 4x^2 - 12x + 14$$
Visible text: 1.
2.
### Answer Key
1. Let's test with the value $$m = 1$$.
- **Left-Hand Side:**
```math
(2m - 3)^3 = (2(1) - 3)^3
```
```math
= (2 - 3)^3
```
```math
= (-1)^3
```
```math
= -1
```
- **Right-Hand Side:**
```math
8m^3 - 27 = 8(1)^3 - 27
```
```math
= 8(1) - 27
```
```math
= 8 - 27
```
```math
= -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 $$B = 3$$.
```math
(2x - 3)^2 + 5 = [(2x)^2 - 2(2x)(3) + (3)^2] + 5
```
```math
= [4x^2 - 12x + 9] + 5
```
```math
= 4x^2 - 12x + 9 + 5
```
```math
= 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**.
Visible text: 1. Let's test with the value .
- **Left-Hand Side:**
- **Right-Hand Side:**
Since for , the left-hand side () the right-hand side (), this equation is **not a polynomial identity**.
The correct identity is .
2. We will expand the left-hand side using the identity , with and .
Since the result of expanding the left-hand side () is exactly the same as the right-hand side, this equation is **proven to be a polynomial identity**.