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The concept of multiplicity of zeros refers to how often a particular zero appears in a polynomial function. In the function \( f(x) = (x-1)^n (x+1)^n \), we have zeros at \( x=1 \) and \( x=-1 \), each with a multiplicity of \( n \). The multiplicity of a zero influences the shape and behavior of the polynomial graph near those zeros.

When a zero has a multiplicity of one, or \( n=1 \), it is called a simple zero. The graph of the polynomial crosses the x-axis at this zero. For our function, with \( n=1 \), the graph will intersect the axis at both \( x=1 \) and \( x=-1 \).

If the multiplicity is two (\( n=2 \)), the zero becomes a double zero. The graph will touch the x-axis at these points but will not cross it. When \( n=3 \), the multiplicity is three, leading to a zero known as a triple zero. Here, the graph flattens more and crosses the axis with a directional shift. As the multiplicity increases to four (\( n=4 \)), the zero is a quadruple zero, resulting in the graph appearing even flatter at those points, making it just touch the x-axis without crossing.

Graphing functions involves visualizing how the polynomial behaves on a coordinate plane. To graph the function \( f(x) = (x-1)^n (x+1)^n \), it is crucial to understand how the multiplicities alter the graph's appearance.

• For \( n=1 \), each zero at \( x=1 \) and \( x=-1 \) causes the graph to pass straight through the x-axis at these points.
• For \( n=2 \), the graph touches the x-axis at these zeros but doesn't cross. This indicates that the graph just "kisses" the axis, showing a peak or trough depending on the direction of approach.
• With \( n=3 \), you observe a more pronounced "flattening" or "turning" effect at the zeros as the graph crosses the axis. It flattens around these points, showing an inflection in direction.
• Finally, for \( n=4 \), the graph becomes extremely flat and wide around \( x=1 \) and \( x=-1 \), creating more significant peaks or troughs that do not intersect the axis.

Overall, graphing is not just about plotting points but understanding how the zeros and their multiplicities affect the overall shape of the graph. It tells us how the function behaves near these critical points.

The behavior of a polynomial function at intercepts is guided by the multiplicity of its zeros. For the function \( f(x) = (x-1)^n (x+1)^n \), we analyze how the graph behaves particularly at \( x=1 \) and \( x=-1 \), which are the intercepts.

• If \( n \) is even, like 2 or 4, the polynomial graph touches the x-axis at these intercepts but doesn't cross. This touching point indicates that the function approaches zero at \( x=1 \) and \( x=-1 \), remains non-negative or non-positive depending on the overall graph direction.
• For odd \( n \), such as 1 or 3, the graph does cross the x-axis at intercepts. The crossing happens with a change of direction, portraying that the function transitions from negative to positive or vice versa through these intercepts.

The behavior at these intercepts provides insight into how the polynomial interacts dynamically with the x-axis based on its multiplicity. Understanding this allows us to predict and interpret the graphical representation of polynomials effectively.