Polynomials: Geometrical Meaning of Zeroes
Visually confirm that the **zeroes** of a quadratic polynomial are the **x-coordinates** of the points where its graph intersects the x-axis.
Help & Instructions
▼The zeroes of a polynomial $P(x)$ are the values of $x$ for which $P(x) = 0$. On a graph of $y=P(x)$, this occurs where $y=0$, which is the x-axis.
- **Define Equation:** Enter coefficients $a$, $b$, and $c$ for the polynomial $y = ax^2 + bx + c$.
- **Plot Curve:** Click "Plot Parabola" to draw the graph.
- **Find Zeroes:** The points where the red curve crosses the horizontal (x-axis) are highlighted. The tool calculates their exact coordinates.
- **Explore Cases:** Try $(1, 0, 0)$ for one zero, or $(1, 0, 4)$ for no real zeroes.
Zeroes of the Polynomial
The **zeroes of a polynomial** are the solutions to $P(x)=0$. For a quadratic, the graph is a parabola. The number of times the parabola crosses the x-axis determines the number of **real zeroes** (0, 1, or 2).
The Quadratic Formula Connection
The zeroes are found using the quadratic formula, where $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The term $b^2 - 4ac$, the **discriminant**, determines the number of zeroes:
- If $\Delta > 0$: Two distinct real zeroes (parabola crosses x-axis twice).
- If $\Delta = 0$: One real zero (parabola touches the x-axis once).
- If $\Delta < 0$: No real zeroes (parabola does not cross x-axis).
Finding the zeroes of quadratic functions is essential in:
- **Physics:** Calculating when a projectile hits the ground (height = 0).
- **Engineering:** Determining optimal points (maxima and minima) for curves.
- **Finance:** Modeling profit functions to find the break-even points (profit = 0).
