lf `p(x) = ax^2 + bx+c`, then `-b/a` is equal to   :

Options

A. 0
B. 1
C. product of zeroes
D. sum of zeroes

Answer

To solve the problem, we need to find out what \(-\frac{b}{a}\) represents in the context of the quadratic polynomial \(p(x) = ax^2 + bx + c\).

### Step-by-Step Solution:

1. **Understanding the Polynomial**:
We start with the quadratic polynomial given by:
\[ p(x) = ax^2 + bx + c \]
Here, \(a\), \(b\), and \(c\) are constants.

**Hint**: Identify the standard form of a quadratic equation.

2. **Finding the Roots**:
The roots (or zeros) of the quadratic polynomial can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula gives us two roots, which we can denote as \(\alpha\) and \(\beta\).

**Hint**: Recall the quadratic formula and how it relates to the roots of the polynomial.

3. **Sum of the Roots**:
The sum of the roots \(\alpha + \beta\) can be calculated as follows:
\[ \alpha + \beta = \frac{-b + \sqrt{b^2 - 4ac}}{2a} + \frac{-b - \sqrt{b^2 - 4ac}}{2a} \]
When we add these two expressions, the square root terms cancel out:
\[ \alpha + \beta = \frac{-b + \sqrt{b^2 - 4ac} - b - \sqrt{b^2 - 4ac}}{2a} = \frac{-2b}{2a} \]

**Hint**: Simplify the expression carefully to see how the terms interact.

4. **Final Simplification**:
Now, simplifying \(\frac{-2b}{2a}\) gives:
\[ \alpha + \beta = \frac{-b}{a} \]
Thus, we conclude that:
\[ -\frac{b}{a} = \text{Sum of the roots} \]

**Hint**: Recognize that the sum of the roots of a quadratic equation is given by \(-\frac{b}{a}\).

### Conclusion:
The expression \(-\frac{b}{a}\) is equal to the sum of the roots (or zeros) of the quadratic polynomial \(p(x) = ax^2 + bx + c\).