Given the function:
$P(r) = (r - 1)(r + 6)(r - 3)$

Step 1: Find the $P$-Intercept

The $P$-intercept occurs when $r = 0$, meaning we substitute $r = 0$ into the function:

$P(0) = (0 - 1)(0 + 6)(0 - 3)$

$= (-1)(6)(-3)$

$= 18$

So, the $P$-intercept is $(0, 18)$.

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Step 2: Find the $r$-Intercepts

The $r$-intercepts occur where $P(r) = 0$, meaning we solve:

$(r - 1)(r + 6)(r - 3) = 0$

Setting each factor equal to zero:

$r - 1 = 0 \quad \Rightarrow \quad r = 1$

$r + 6 = 0 \quad \Rightarrow \quad r = -6$

$r - 3 = 0 \quad \Rightarrow \quad r = 3$

So, the $r$-intercepts are $(-6,0), (1,0), (3,0)$.

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Related Questions:

1. What does the sign of the leading coefficient tell us about the end behavior of $P(r)$?
2. How can you determine whether the function has a local maximum or minimum?
3. What happens if an additional factor $(r + 2)$ is added to the function?
4. Can this function be rewritten in expanded polynomial form?
5. How do you determine the multiplicity of an $r$-intercept?

Tip:

The product of all the $r$-intercepts gives the constant term in the polynomial when expanded.