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To understand the concept of zeros, it's essential to recognize what they represent in the context of a function. The zeros of a function are the x-values where the function equals zero. In simpler terms, these are the points where the graph of the function crosses or touches the x-axis. To find these zeros in the given function, we have to solve the equation by setting each factor to zero. Each solution to these equations indicates a zero of the function. For example, in our original function, the factors are \( x^2 \), \((2x+3)^5\), and \((x-4)^2\). Setting each factor to zero, we find:\[ \begin{align*} 1. & \quad x^2 = 0 \Rightarrow x = 0, \2. & \quad (2x+3)^5 = 0 \Rightarrow 2x + 3 = 0 \Rightarrow x = -\frac{3}{2}, \3. & \quad (x-4)^2 = 0 \Rightarrow x - 4 = 0 \Rightarrow x = 4. \end{align*} \]These values \(0, -\frac{3}{2},\) and \(4\) are the zeros of the function, signifying the x-values where the function becomes zero.

Factorization is the process of breaking down an expression into a product of simpler factors. In polynomial functions, factorizing is an essential step to easily identify the zeros.For the function \( f(x) = x^{2}(2x+3)^{5}(x-4)^{2} \), the expression is already factorized into three distinct parts: \( x^2 \), \((2x+3)^5\), and \((x-4)^2\). These factors are simpler expressions that, when multiplied together, give the original function.Identifying these factors is crucial because each factor represents a potential zero. Specifically, each term in a factorized expression has a role in determining the fundamental traits of the function, like its zeros and their characteristics. Hence, breaking down complex polynomial expressions into factors is a fundamental step in analyzing and understanding polynomial functions.

Multiplicity refers to the number of times a specific zero appears in a polynomial function. It is dictated by the exponent on the factor associated with that zero. The notion of multiplicity provides more insight into the behavior of a function around its zeros.For instance, if a zero has a multiplicity of 1, the graph of the function will cross the x-axis at that point. However, if the multiplicity is greater than 1, like 2 or 5 as in our function, the graph will touch but not necessarily cross the x-axis, creating a "bounce."In our exercise, we determined the multiplicities as follows:

• For zero \( x = 0 \), associated with the factor \( x^2 \), the multiplicity is 2.
• For zero \( x = -\frac{3}{2} \), associated with the factor \( (2x+3)^5 \), the multiplicity is 5.
• For zero \( x = 4 \), associated with the factor \( (x-4)^2 \), the multiplicity is 2.

Understanding multiplicity helps predict how the graph behaves at those zeros. Higher multiplicities indicate stronger "bumping" or "bouncing" behavior at the point where the function becomes zero. This aspect of analysis makes it easier to sketch the graph of polynomial functions accurately.