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Understanding the zeros of a polynomial function and their multiplicities is crucial for graphing. When we say a function has a "zero," we mean there is some value of \(x\) that makes the function equal zero. For the polynomial \(f(x) = -5x^4 + 10x^3 - 5x^2\), we factor it as \(-5x^2(x-1)^2\) to find the zeros. This gives us zeros at \(x = 0\) and \(x = 1\).
Both zeros here have a multiplicity of 2, indicating that each zero appears twice in the factorization.

• Zero at \(x = 0\) with multiplicity 2
• Zero at \(x = 1\) with multiplicity 2

The multiplicity helps us understand how the graph behaves near these zeros. A zero with even multiplicity means the graph will "touch" the x-axis at this point but will not cross it. This is a key feature to recognize when sketching polynomial graphs.

The behavior of a polynomial graph at its intercepts provides information about its shape and structure. At each zero, the graph behaves differently depending on the multiplicity:

• At \(x = 0\) with even multiplicity (2), the graph touches the x-axis and turns back without crossing it.
• At \(x = 1\) with even multiplicity (2), the graph likewise touches and turns back without crossing.

This "touch and return" behavior means that at both \(x = 0\) and \(x = 1\), the graph does not switch sides of the x-axis.
To understand further graph behavior, we evaluate some points:

• At \(x = -1\), the graph is below the x-axis with a value of -5.
• At \(x = 0.5\), the graph remains slightly below the x-axis, at approximately -0.3125.
• At \(x = 2\), it is far below the x-axis, at -20.

These points help identify the precise shape of the graph between zeros.

The end behavior of a polynomial function is driven by its leading term. Here, the leading term is \(-5x^4\), which helps predict how the polynomial behaves as \(x\) approaches infinity or negative infinity.
Because the coefficient is negative and the degree is even, the graph of \(f(x)\) will head downwards towards \(-\infty\) as \(x\) moves to \(\pm \infty\).

• As \(x \to \infty\), \(f(x) \to -\infty\).
• As \(x \to -\infty\), \(f(x) \to -\infty\).

This symmetry due to the even degree tells us that both ends of the graph will have the same direction. They mirror each other heading consistently towards negative infinity.

Intercepts are vital for plotting and interpreting graphs. The x-intercepts of a function occur where the function equals zero. For \(f(x) = -5x^2(x-1)^2\), these intercepts are:\(-x = 0\) and \(x = 1\), with each intercept having a multiplicity of 2, meaning it touches but does not cross the x-axis at these points.
The y-intercept occurs where the graph crosses the y-axis, that is, when \(x = 0\).
By plugging \(x = 0\) into the function, \(f(0) = -5(0)^2(0 - 1)^2 = 0\), we find that the y-intercept is at the origin (0,0).

• X-Intercepts: \(x = 0\) (multiplicity 2), \(x = 1\) (multiplicity 2)
• Y-Intercept: (0, 0)

These intercepts provide anchor points from which we plot and draw out the polynomial's graph.