Find the zeroes of the quadratic polynomial `x^(2) + 5x + 6` and verify the relationship between the zeroes and the coefficients.

Options

A. `-3, 3`
B. `-3, 2`
C. `-3, -3`
D. `-3, -2`

Answer

To find the zeroes of the quadratic polynomial \( x^2 + 5x + 6 \) and verify the relationship between the zeroes and the coefficients, we will follow these steps:

### Step 1: Write the quadratic polynomial
The given quadratic polynomial is:
\[ f(x) = x^2 + 5x + 6 \]

### Step 2: Factor the quadratic polynomial
To find the zeroes, we need to factor the polynomial. We look for two numbers that multiply to \( 6 \) (the constant term) and add up to \( 5 \) (the coefficient of \( x \)). The numbers \( 2 \) and \( 3 \) satisfy these conditions because:
\[ 2 \times 3 = 6 \quad \text{and} \quad 2 + 3 = 5 \]
Thus, we can factor the polynomial as:
\[ f(x) = (x + 2)(x + 3) \]

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

### Step 4: Solve for \( x \)
This gives us two equations:
1. \( x + 2 = 0 \) → \( x = -2 \)
2. \( x + 3 = 0 \) → \( x = -3 \)

Thus, the zeroes of the polynomial are:
\[ x = -2 \quad \text{and} \quad x = -3 \]

### Step 5: Verify the relationship between the zeroes and the coefficients
The sum of the zeroes \( \alpha \) and \( \beta \) is:
\[ \alpha + \beta = -2 + (-3) = -5 \]
According to the relationship, the sum of the zeroes is given by:
\[ -\frac{b}{a} \]
where \( b = 5 \) and \( a = 1 \) (from the polynomial \( ax^2 + bx + c \)). Thus:
\[ -\frac{b}{a} = -\frac{5}{1} = -5 \]
This confirms that the sum of the zeroes is equal to \( -\frac{b}{a} \).

Next, we calculate the product of the zeroes:
\[ \alpha \cdot \beta = (-2) \cdot (-3) = 6 \]
According to the relationship, the product of the zeroes is given by:
\[ \frac{c}{a} \]
where \( c = 6 \) and \( a = 1 \). Thus:
\[ \frac{c}{a} = \frac{6}{1} = 6 \]
This confirms that the product of the zeroes is equal to \( \frac{c}{a} \).

### Conclusion
The zeroes of the polynomial \( x^2 + 5x + 6 \) are \( -2 \) and \( -3 \). We have verified that the sum and product of the zeroes satisfy the relationships with the coefficients.

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