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The degree of a polynomial is crucial in understanding the overall shape of its graph. In the polynomial function given as \( P(x) = x(x-3)(x+2) \), the degree is determined by the highest power of \( x \) when the expression is fully expanded. Let's break it down: multiplying out the terms, we have \( x (x-3) (x+2) = x^3 - x^2 - 6x \).

This results in the highest power of \( x \) being 3, which means the function is a cubic polynomial. A degree of 3 often indicates the graph will cross the x-axis at three different points and exhibit a particular cubic-like shape, featuring either one hill and one valley or potentially more intricate curves depending on additional polynomial terms.

X-intercepts are the points where the graph meets the x-axis. These are fundamental for sketching the graph and understanding the behavior of the function. To find them in the polynomial \( P(x) = x(x-3)(x+2) \), we set the polynomial function equal to zero:

\( x(x-3)(x+2) = 0 \).

We solve for \( x \) by setting each factor of the polynomial to zero individually:

• From \( x = 0 \), we have an intercept at (0, 0).
• From \( x-3 = 0 \), we get \( x = 3 \), which means there is an intercept at (3, 0).
• From \( x+2 = 0 \), we find \( x = -2 \), providing an intercept at (-2, 0).

These intercepts are the points where the polynomial touches or crosses the x-axis, informing the placement of key points when sketching the graph.

The end behavior of a polynomial function reveals how the graph acts as \( x \) approaches positive or negative infinity. This is governed by the leading term, which, for the polynomial \( P(x) = x^3 - x^2 - 6x \), is \( x^3 \).

Since the leading coefficient of \( x^3 \) is positive, the graph will behave in the following manner:

• As \( x \rightarrow +fty \), \( P(x) \rightarrow +fty \).
• As \( x \rightarrow -fty \), \( P(x) \rightarrow -fty \).

This indicates that the graph rises on the right side and falls on the left side, typical of an odd degree polynomial with a positive leading coefficient. Understanding this behavior is essential for correctly sketching the overall graph.

The y-intercept of a polynomial function tells us where the graph crosses the y-axis. It occurs when \( x = 0 \). For the function \( P(x) = x(x-3)(x+2) \), we calculate the y-intercept by evaluating the function at \( x = 0 \):

\( P(0) = 0(0-3)(0+2) = 0 \).

Thus, the y-intercept is located at (0, 0).

In this particular case, it is noteworthy that the y-intercept coincides with one of the x-intercepts, indicating that the graph crosses both the x and y-axes at the origin. This point is important as it confirms that the origin is a critical point where the function intersects both axes.