- Identify the form of the polynomial as \(f(x) = a(x - r)^2\) using the given root and multiplicity.
- Substitute the leading coefficient \(a = -1\) and root \(r = 4\).
- Expand the expression: \(f(x) = -1(x - 4)^2 = -1(x^2 - 8x + 16) = -x^2 + 8x - 16\).
- The correct polynomial is \(\boxed{f(x) = -x^2 + 8x - 16}\).
Explanation
- Analysis of the Polynomial Function Conditions
We are looking for a second-degree polynomial function \(f(x)\) given a specific leading coefficient and a root with its multiplicity. A second-degree polynomial (quadratic function) can generally be expressed in the form \(f(x) = a(x - x_1)(x - x_2)\), where \(a\) is the leading coefficient and \(x_1, x_2\) are the roots of the function. If a polynomial has a root \(r\) with multiplicity \(2\), this means that the factor \((x - r)\) appears twice in its factorization. Therefore, the function can be written as \(f(x) = a(x - r)^2\). In this problem, we are given that the leading coefficient is \(a = -1\) and the root is \(r = 4\) with multiplicity \(2\).
- Expanding and Identifying the Polynomial Function
Using the form \(f(x) = a(x - r)^2\), we substitute the given values \(a = -1\) and \(r = 4\):
\[f(x) = -1 \cdot (x - 4)^2\]
To find the standard form of the polynomial, we expand the squared binomial \((x - 4)^2\):
\[(x - 4)^2 = (x - 4)(x - 4) = x^2 - 4x - 4x + 16 = x^2 - 8x + 16\]
Now, we multiply this expansion by the leading coefficient \(-1\):
\[f(x) = -1 \cdot (x^2 - 8x + 16)\]
\[f(x) = -x^2 + 8x - 16\]
Comparing this result with the given choices, we see that it matches the second option.
Examples
Understanding how to construct a polynomial from its roots is fundamental in fields such as engineering and physics, where scientists often model physical phenomena like the trajectory of a projectile. By knowing the points where a path crosses an axis (the roots) and the curvature (governed by the leading coefficient), they can construct a mathematical model to predict the object's height at any given time. For example, if a path is modeled by a quadratic function, knowing the vertex or the roots allows engineers to optimize the design of structures or analyze the dynamics of moving objects effectively.