Which form of a cubic function indicates it has a double root?
A cubic function with a double root is indicated by the fact that it touches the x-axis at a single point without crossing or intersecting it. Mathematically, this means that the cubic function has a factor that appears twice in its factored form.
For example, if a cubic equation is given in the form f(x) = (x - r)(x - r)(x - s), where r and s are the roots of the equation, then the double root is indicated by the squared factor (x - r)(x - r).
To determine which form of a cubic function indicates it has a double root, we need to understand the concept of roots in general and how they are represented in the equation of a cubic function.
A root, also known as a zero or x-intercept, is a value for which a function evaluates to zero. In a cubic function, there can be three distinct roots (where the cubic function crosses the x-axis at three different points) or fewer if there are repeated or shared roots.
In general, a cubic function can be expressed in the form:
f(x) = ax^3 + bx^2 + cx + d
To indicate a double root in a cubic function, the double root must occur when the function touches or rebounds from the x-axis at the same point. In terms of the cubic function equation, this can be identified by having a factor that appears twice in the factored form of the function.
The factored form of a cubic function is written as:
f(x) = a(x - r)(x - s)(x - t)
where r, s, and t are the roots of the function, and 'a' is a constant.
If a cubic function has a double root, it means that two of the factors in the factored form are the same. Therefore, the form that indicates a double root is:
f(x) = a(x - r)^2(x - t)
In this form, (x - r) appears as a repeated factor, indicating that 'r' is a double root.
In summary, if a cubic function is written in the form f(x) = a(x - r)^2(x - t), where (x - r) is a repeated factor, then it indicates that the function has a double root at 'r'.