Let's go through this problem step-by-step.

1. Finding the Roots

Given the function: $f(x) = 7(x + 6)(x + 4)(x - 7)$ we can see that $f(x)$ is in factored form, and we can identify the roots by setting each factor to zero:

• $x + 6 = 0 \Rightarrow x = -6$
• $x + 4 = 0 \Rightarrow x = -4$
• $x - 7 = 0 \Rightarrow x = 7$

Thus, the roots are $x = -6$, $x = -4$, and $x = 7$, listed in increasing order.

2. Analyzing the Sign of $f(x)$ Around the Roots

To determine where the graph of $f(x)$ is above or below the x-axis, we examine the intervals around the roots:

• To the left of the first root $x = -6$: Choose a point, say $x = -7$. Substituting into $f(x)$: $f(-7) = 7(-7 + 6)(-7 + 4)(-7 - 7) = 7(-1)(-3)(-14) < 0$ This calculation implies that $f(x)$ is below the x-axis to the left of $x = -6$.

• Between the first two roots, $x = -6$ and $x = -4$: Choose a point, say $x = -5$. Substituting into $f(x)$: $f(-5) = 7(-5 + 6)(-5 + 4)(-5 - 7) = 7(1)(-1)(-12) > 0$ This calculation implies that $f(x)$ is above the x-axis between $x = -6$ and $x = -4$.

• Between the last two roots, $x = -4$ and $x = 7$: Choose a point, say $x = 0$. Substituting into $f(x)$: $f(0) = 7(0 + 6)(0 + 4)(0 - 7) = 7(6)(4)(-7) < 0$ This calculation implies that $f(x)$ is below the x-axis between $x = -4$ and $x = 7$.

• After the last root $x = 7$: Choose a point, say $x = 8$. Substituting into $f(x)$: $f(8) = 7(8 + 6)(8 + 4)(8 - 7) = 7(14)(12)(1) > 0$ This calculation implies that $f(x)$ is above the x-axis after $x = 7$.

Summary of Solutions:

• The roots are $-6$, $-4$, and $7$.
• The sign of $f(x)$ around these roots is as follows:
  • To the left of $x = -6$: below
  • Between $x = -6$ and $x = -4$: above
  • Between $x = -4$ and $x = 7$: below
  • After $x = 7$: above

Would you like further details or have any questions?

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Additional Questions:

1. How does the factor $7$ affect the graph's shape and scale?
2. Why does the graph change direction at each root?
3. How would adding an additional factor change the number of roots and sign changes?
4. How can you determine end behavior for a polynomial function?
5. What would happen to the graph if the coefficient $7$ were negative?

Tip:

The sign of a polynomial function between roots depends on whether each interval includes an odd or even number of root-crossings, which helps determine if the function is positive or negative in those intervals.