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When solving polynomial equations, one of the first steps is often factoring the polynomial. Factoring means breaking down a polynomial into simpler expressions called factors that, when multiplied together, give the original polynomial. For example, the polynomial \( x^2 - 1 \) can be factored as \( (x - 1)(x + 1) \), because multiplying \( (x - 1) \) and \( (x + 1) \) returns \( x^2 - 1 \). Another polynomial, \( x^2 + 4x + 3 \), factors to \( (x + 1)(x + 3) \). Each factor like \( x - 1 \) or \( x + 3 \) can further reveal important properties about the polynomial, such as its zeros and their multiplicities. Understanding how to factor polynomials is crucial for solving polynomial equations and finding real zeros efficiently.

Multiplicity refers to the number of times a particular zero appears as a root of the polynomial. In the given example, we had to find the zeros of the function \( f(x) = 3 (x - 1)(x + 1)(x + 1)^2 (x + 3)^2 \). Each factor contributes to the final count of zeros. For instance, the zero \( x = 1 \) appears only once, giving it a multiplicity of 1. The zero \( x = -1 \) shows up three times: once as \( x + 1 \) and twice as \( (x + 1)^2 \), thus it has a multiplicity of 3. Similarly, \( x = -3 \) is counted twice due to the \( (x + 3)^2 \) term, making its multiplicity 2. Recognizing multiplicities is vital because it affects the behavior of the polynomial function at those points.

To solve a polynomial equation, we need to set the polynomial equal to zero. This reveals where the polynomial crosses the x-axis, or its roots/zeros. From the previous steps, we have the function \( f(x) = 3 (x - 1)(x + 1)(x + 1)^2 (x + 3)^2 \). To find the zeros, we set the function equal to zero and solve: \[ 3 (x - 1)(x + 1)(x + 1)^2 (x + 3)^2 = 0 \]. This equation implies we need to solve each factor for zero: \( x - 1 = 0 \), \( x + 1 = 0 \), and \( x + 3 = 0 \). Solving these gives \( x = 1 \), \( x = -1 \), and \( x = -3 \), which are the roots of the polynomial. Solving polynomial equations often involves factoring, setting each factor to zero, and solving these simpler equations.

Real zeros of a function are the values of \( x \) which make the function equal to zero. These are the x-values where the graph of the function touches or crosses the x-axis. For our example function \( f(x) = 3 (x - 1)(x + 1)(x + 1)^2 (x + 3)^2 \), the real zeros can be found by solving \( 3 (x - 1)(x + 1)(x + 1)^2 (x + 3)^2 = 0 \). This reveals that the real zeros of the function are \( x = 1 \), \( x = -1 \), and \( x = -3 \). The graph of this function touches the x-axis at these points. Finding real zeros is essential, as these points provide key information about the behavior and roots of the polynomial equation.