To determine the missing factor of the given polynomial, let's start by analyzing the provided factor and understanding the structure of the given 3rd degree polynomial.
We are given one factor:
[tex]\[ x^2 + 2x + 4 \][/tex]
Since the polynomial is of 3rd degree, it must have three roots in total. The quadratic factor provided accounts for two of these roots. Therefore, there must be one additional linear factor.
Let the polynomial be [tex]\( P(x) \)[/tex] and assume it can be written as:
[tex]\[ P(x) = (x - a)(x^2 + 2x + 4) \][/tex]
where [tex]\( (x - a) \)[/tex] is the missing linear factor, and [tex]\( a \)[/tex] is the root of this linear factor.
To identify [tex]\( a \)[/tex], we should observe the graph of the polynomial to determine its root. Since the graph is not provided here, we will derive it theoretically by understanding that the polynomial crosses the x-axis at [tex]\( x = a \)[/tex].
For the purpose of illustration, let's assume that from the graph, the polynomial intersects the x-axis at [tex]\( x = 1 \)[/tex]. This assumption helps us form the missing linear factor as follows:
[tex]\[ x - a = x - 1 \][/tex]
Thus, the missing factor is:
[tex]\[ \boxed{x - 1} \][/tex]
If there were another value observed from the graph, you would replace [tex]\( 1 \)[/tex] with that specific value. Note that interpreting the graph correctly is crucial in determining the exact root for the linear factor.
