The given function is:

$P(x) = x(x - 8)(x + 7)$

Key features of this polynomial:

1. Degree of the polynomial: The degree is 3, as it is the product of three linear factors $x$, $(x-8)$, and $(x+7)$.
2. Roots of the polynomial:
   • $x = 0$ (from the factor $x$)
   • $x = 8$ (from the factor $x - 8$)
   • $x = -7$ (from the factor $x + 7$)
3. End behavior:
   • As $x \to \infty$, $P(x) \to \infty$, since the leading term is $x^3$.
   • As $x \to -\infty$, $P(x) \to -\infty$.
4. Y-intercept:
   • The y-intercept is the value of $P(0)$: $P(0) = 0(0 - 8)(0 + 7) = 0$ Hence, the y-intercept is at $(0, 0)$.
5. Intervals of increase and decrease:
   • To determine where the function is increasing or decreasing, we calculate the first derivative $P'(x)$.

First Derivative of $P(x)$:

Using the product rule:

$P(x) = x(x-8)(x+7)$

Let $u = x$, $v = (x-8)(x+7)$. Then,

$P'(x) = u'v + uv'$

Step 1: Differentiate $u$:

$u = x \implies u' = 1$

Step 2: Differentiate $v$:

$v = (x - 8)(x + 7)$ Using the product rule: $v' = (x-8)'(x+7) + (x-8)(x+7)'$ $v' = 1(x+7) + (x-8)(1) = (x+7) + (x-8) = 2x - 1$

Step 3: Combine results:

$P'(x) = (1)((x-8)(x+7)) + x(2x-1)$ $P'(x) = (x-8)(x+7) + x(2x-1)$

Expand and simplify $P'(x)$:

$(x-8)(x+7) = x^2 - x - 56$ $P'(x) = (x^2 - x - 56) + (2x^2 - x)$ $P'(x) = 3x^2 - 2x - 56$

Critical Points:

To find the critical points, solve $P'(x) = 0$: $3x^2 - 2x - 56 = 0$

This is a quadratic equation. Solve using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 3$, $b = -2$, $c = -56$: $x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(-56)}}{2(3)}$ $x = \frac{2 \pm \sqrt{4 + 672}}{6}$ $x = \frac{2 \pm \sqrt{676}}{6}$ $x = \frac{2 \pm 26}{6}$

Solutions: $x = \frac{2 + 26}{6} = 4.67, \quad x = \frac{2 - 26}{6} = -4$

Summary of critical points:

• $x = -4$
• $x \approx 4.67$

These divide the x-axis into intervals for testing $P'(x)$.

Would you like further analysis, such as the intervals of increase/decrease or a sketch of the graph?

Questions:

1. How would the behavior of $P(x)$ change if the degree were even instead of odd?
2. What are the implications of a triple root in a polynomial function?
3. How does the y-intercept change if a constant is added to $P(x)$?
4. Can you determine the maximum/minimum values using the second derivative test?
5. What is the relationship between the signs of the roots and the graph's orientation?

Tip:

Always check the multiplicity of roots—odd multiplicities indicate the graph crosses the x-axis, while even multiplicities mean it just touches the x-axis.