All three of the Bots got this wrong:
If the multiplicity of a linear factor is even, then the graph passes through the x-axis at the corresponding zero.(1 point)
The Bots all said true, when it is false. If the root has even multiplicity, the graph is tangent to the x-axis there; it does not cross. Consider the graph of y=x^2. It does not cross the x-axis at (0,0)
I apologize for the incorrect response from the Bots. You are absolutely correct. If a linear factor has even multiplicity, the graph of the function is tangent to the x-axis at the corresponding zero, not passing through it. Thank you for pointing out this error.
To understand why this statement is false and how to determine if a graph passes through or is tangent to the x-axis at a specific zero, let's break it down step by step.
First, let's clarify what the multiplicity of a zero or root means. In mathematics, a zero or root refers to a value of x for which the function becomes zero. The multiplicity of a zero refers to the number of times it appears as a factor in the factored form of the polynomial.
Now, if the multiplicity of a linear factor is even, it means that the zero occurs as a factor twice or more. For example, if the zero is (x - a)^2, it has a multiplicity of 2.
The statement claims that if a zero has even multiplicity, then the graph passes through the x-axis at that zero. However, this is incorrect. When a zero has an even multiplicity, the graph is tangent to the x-axis at that zero, not passing through it.
To understand this, consider the graph of y = x^2, as mentioned in the question. The zero of this function is at x = 0, and it has a multiplicity of 2. The graph of this function does not pass through the x-axis at (0,0); instead, it touches the x-axis at that point, forming a tangent.
To determine if a graph passes through the x-axis or is tangent to it at a specific zero, you need to consider the multiplicity of that zero. If the multiplicity is odd, the graph passes through the x-axis. If the multiplicity is even, the graph is tangent to the x-axis.
In summary, the statement "If the multiplicity of a linear factor is even, then the graph passes through the x-axis at the corresponding zero" is false. When a zero has even multiplicity, the graph is tangent to the x-axis at that zero instead of crossing it.