Find zeroes of the polynomial `x^(2)-4` and verify the relation between zeroes and coefficients.

Explanation

To find the zeroes of the polynomial \( p(x) = x^2 - 4 \) and verify the relation between zeroes and coefficients, we can follow these steps:

### Step 1: Identify the polynomial
We start with the polynomial:
\[ p(x) = x^2 - 4 \]

### Step 2: Factor the polynomial
We can recognize that \( x^2 - 4 \) is a difference of squares, which can be factored as:
\[ p(x) = (x - 2)(x + 2) \]

### Step 3: Set the polynomial equal to zero
To find the zeroes, we set the polynomial equal to zero:
\[ (x - 2)(x + 2) = 0 \]

### Step 4: Solve for \( x \)
We can solve this equation by setting each factor equal to zero:
1. \( x - 2 = 0 \) → \( x = 2 \)
2. \( x + 2 = 0 \) → \( x = -2 \)

Thus, the zeroes of the polynomial are:
\[ x = 2 \quad \text{and} \quad x = -2 \]

### Step 5: Verify the relation between zeroes and coefficients
The polynomial can be expressed in the standard form \( ax^2 + bx + c \):
- Here, \( a = 1 \), \( b = 0 \), and \( c = -4 \).

#### Relation 1: Sum of the roots
The sum of the roots \( (r_1 + r_2) \) should equal \( -\frac{b}{a} \):
\[ r_1 + r_2 = 2 + (-2) = 0 \]
\[ -\frac{b}{a} = -\frac{0}{1} = 0 \]
Both sides are equal, confirming the relation.

#### Relation 2: Product of the roots
The product of the roots \( (r_1 \cdot r_2) \) should equal \( \frac{c}{a} \):
\[ r_1 \cdot r_2 = 2 \cdot (-2) = -4 \]
\[ \frac{c}{a} = \frac{-4}{1} = -4 \]
Both sides are equal, confirming this relation as well.

### Conclusion
The zeroes of the polynomial \( x^2 - 4 \) are \( 2 \) and \( -2 \), and the relations between the zeroes and coefficients have been verified.

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