To find the $y$-intercept and $x$-intercepts of the function $P(x) = (x - 6)(x + 5)(8x - 3)$, we can proceed as follows:

1. Finding the $y$-intercept

The $y$-intercept occurs where $x = 0$. We substitute $x = 0$ into $P(x)$:

$P(0) = (0 - 6)(0 + 5)(8 \cdot 0 - 3) = (-6)(5)(-3) = 90$

So, the $y$-intercept is: $(0, 90)$

2. Finding the $x$-intercepts

The $x$-intercepts occur where $P(x) = 0$. This is the case when any factor of $P(x)$ is equal to zero:

$(x - 6) = 0 \Rightarrow x = 6$ $(x + 5) = 0 \Rightarrow x = -5$ $(8x - 3) = 0 \Rightarrow x = \frac{3}{8}$

So, the $x$-intercepts are: $(6, 0), \, (-5, 0), \, \text{and} \, \left(\frac{3}{8}, 0\right)$

Summary

• $y$-intercept: $(0, 90)$
• $x$-intercepts: $(6, 0), (-5, 0), \left(\frac{3}{8}, 0\right)$

Would you like more details, or do you have any questions?

Further Questions:

1. How would you determine the symmetry of the function $P(x)$?
2. Can you describe the end behavior of $P(x)$ as $x \to \pm \infty$?
3. What would the graph of $P(x)$ look like around each intercept?
4. How can you confirm the multiplicity of each $x$-intercept?
5. What role do each of the factors play in the overall shape of $P(x)$'s graph?

Tip: For polynomial functions, $x$-intercepts correspond to the roots, and the $y$-intercept is found by evaluating the function at $x = 0$.