To find a quadratic polynomial when its zeros are given, you can utilize the fundamental relationship between the roots (zeros) and the coefficients of a quadratic equation. This involves calculating the sum and product of the given zeros and then substituting these values into a standard formula.
What Are Zeros of a Quadratic Polynomial?
The zeros (also known as roots) of a quadratic polynomial are the values of the variable (commonly x) for which the polynomial evaluates to zero. Graphically, these are the x-intercepts of the parabola represented by the quadratic polynomial. A quadratic polynomial typically has two zeros, which can be real or complex.
A general quadratic polynomial can be expressed in the form:
ax² + bx + c = 0
Where a, b, and c are constants and a ≠ 0. If α (alpha) and β (beta) are the zeros of this polynomial, then they satisfy the following relationships with the coefficients:
- Sum of Zeros:
α + β = -b/a - Product of Zeros:
αβ = c/a
These relationships lead directly to the method for constructing the polynomial.
Steps to Construct a Quadratic Polynomial from Its Zeros
Follow these straightforward steps to determine the quadratic polynomial when its zeros are known:
-
Find the Sum of the Zeros
Add the two given zeros together. If the zeros are
αandβ, their sum isα + β. -
Find the Product of the Zeros
Multiply the two given zeros together. The product will be
αβ. -
Substitute into the Formula
Use the standard expression for a quadratic polynomial derived from its zeros:
x² - (Sum of Zeros)x + (Product of Zeros)Substitute the calculated sum and product into this expression to get the required quadratic polynomial.
Here's a quick reference table summarizing these steps:
| Step | Description | Formula/Operation |
|---|---|---|
| 1 | Calculate the Sum | α + β |
| 2 | Calculate the Product | αβ |
| 3 | Form the Polynomial | x² - (α + β)x + (αβ) |
Example: Finding a Quadratic Polynomial
Let's illustrate the process with an example. Suppose the given zeros of a quadratic polynomial are 2 and -5.
-
Find the Sum of Zeros:
Sum = 2 + (-5) = -3 -
Find the Product of Zeros:
Product = 2 * (-5) = -10 -
Substitute into the Formula:
Using the formulax² - (Sum of Zeros)x + (Product of Zeros):
x² - (-3)x + (-10)
x² + 3x - 10
Therefore, the quadratic polynomial whose zeros are 2 and -5 is x² + 3x - 10.
Understanding the General Form and Multiples
The formula x² - (Sum of Zeros)x + (Product of Zeros) provides a quadratic polynomial. However, it's important to note that if α and β are the zeros of a polynomial, then any polynomial of the form k(x - α)(x - β) where k is any non-zero constant, will also have α and β as its zeros.
Expanding k(x - α)(x - β) gives:
k(x² - αx - βx + αβ)
k(x² - (α + β)x + αβ)
This means there are infinitely many quadratic polynomials that share the same zeros, differing only by a constant multiplicative factor k. Unless a specific condition (like a specific leading coefficient or a point the polynomial must pass through) is given, the simplest form (where k=1) is generally the one derived using the steps above.
