Find zeroes of the polynormial `2x^(2)-5x+2` and verify the relation between zeroes and coefficients.

Explanation

To find the zeroes of the polynomial \(2x^2 - 5x + 2\) and verify the relation between the zeroes and coefficients, follow these steps:

### Step 1: Write the polynomial
Let \(p(x) = 2x^2 - 5x + 2\).

### Step 2: Factor the polynomial
We will factor the polynomial by splitting the middle term. We need two numbers that multiply to \(2 \times 2 = 4\) (the product of the coefficient of \(x^2\) and the constant term) and add up to \(-5\) (the coefficient of \(x\)).

The numbers that satisfy this are \(-4\) and \(-1\). Thus, we can rewrite the polynomial as:
\[ p(x) = 2x^2 - 4x - x + 2 \]

### Step 3: Group the terms
Now, we will group the terms:
\[ p(x) = (2x^2 - 4x) + (-x + 2) \]

### Step 4: Factor by grouping
Now, we factor out the common terms from each group:
\[ = 2x(x - 2) - 1(x - 2) \]
\[ = (2x - 1)(x - 2) \]

### Step 5: Set the polynomial equal to zero
To find the zeroes, we set the factored form equal to zero:
\[ (2x - 1)(x - 2) = 0 \]

### Step 6: Solve for \(x\)
This gives us two equations to solve:
1. \(2x - 1 = 0\)
2. \(x - 2 = 0\)

From the first equation:
\[ 2x = 1 \implies x = \frac{1}{2} \]

From the second equation:
\[ x = 2 \]

Thus, the zeroes of the polynomial are \(x = 2\) and \(x = \frac{1}{2}\).

### Step 7: Verify the relation between zeroes and coefficients
We will verify the relations:
1. **Sum of the roots** \( = -\frac{b}{a} \)
2. **Product of the roots** \( = \frac{c}{a} \)

From the polynomial \(2x^2 - 5x + 2\):
- Coefficient \(a = 2\)
- Coefficient \(b = -5\)
- Coefficient \(c = 2\)

#### Sum of the roots:
\[ \text{Sum} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \]
\[ -\frac{b}{a} = -\frac{-5}{2} = \frac{5}{2} \]

Both sums are equal, thus verified.

#### Product of the roots:
\[ \text{Product} = 2 \times \frac{1}{2} = 1 \]
\[ \frac{c}{a} = \frac{2}{2} = 1 \]

Both products are equal, thus verified.

### Conclusion
The zeroes of the polynomial \(2x^2 - 5x + 2\) are \(x = 2\) and \(x = \frac{1}{2}\). The relations between the zeroes and coefficients are verified.

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