Find zeroes of the given quadratic polynomials and verify the relation between zeroes and coefficients :
`x^(2)-6x-16`

Explanation

To find the zeroes of the quadratic polynomial \( p(x) = x^2 - 6x - 16 \) and verify the relation between the zeroes and coefficients, we can follow these steps:

### Step 1: Identify the coefficients
The given polynomial is in the standard form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = -6 \)
- \( c = -16 \)

### Step 2: Use the middle-term splitting method
We need to factor the quadratic polynomial. We look for two numbers that multiply to \( ac = 1 \times (-16) = -16 \) and add to \( b = -6 \).

The two numbers that satisfy these conditions are \( -8 \) and \( 2 \) because:
- \( -8 \times 2 = -16 \)
- \( -8 + 2 = -6 \)

### Step 3: Rewrite the polynomial
We can rewrite the polynomial using these numbers:
\[ p(x) = x^2 - 8x + 2x - 16 \]

### Step 4: Factor by grouping
Now, we group the terms:
\[ p(x) = (x^2 - 8x) + (2x - 16) \]
Factoring out the common terms:
\[ p(x) = x(x - 8) + 2(x - 8) \]
Now, factor out \( (x - 8) \):
\[ p(x) = (x - 8)(x + 2) \]

### Step 5: Find the zeroes
To find the zeroes, we set the factors equal to zero:
1. \( x - 8 = 0 \) → \( x = 8 \)
2. \( x + 2 = 0 \) → \( x = -2 \)

Thus, the zeroes of the polynomial are \( \alpha = -2 \) and \( \beta = 8 \).

### Step 6: Verify the relation between zeroes and coefficients
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

Calculating the sum:
\[ \alpha + \beta = -2 + 8 = 6 \]
Now, calculate \( -\frac{b}{a} \):
\[ -\frac{-6}{1} = 6 \]
Both values match, hence verified.

Calculating the product:
\[ \alpha \beta = (-2) \times 8 = -16 \]
Now, calculate \( \frac{c}{a} \):
\[ \frac{-16}{1} = -16 \]
Both values match, hence verified.

### Conclusion
The zeroes of the polynomial \( x^2 - 6x - 16 \) are \( -2 \) and \( 8 \). The relations \( \alpha + \beta = 6 \) and \( \alpha \beta = -16 \) are verified with the coefficients of the polynomial.

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