Let's analyze the polynomial $p(x) = -3x^4 + 2x^3 - 7^6 + x^5$.

Step-by-step breakdown:

1. Simplify the polynomial: $p(x) = -3x^4 + 2x^3 - 7^6 + x^5$ Notice that $7^6$ is just a constant, which equals $117649$, so: $p(x) = -3x^4 + 2x^3 - 117649 + x^5$ Rearranging the terms: $p(x) = x^5 - 3x^4 + 2x^3 - 117649$

2. Examine each statement:

   a) Constant term: The constant term is $-117649$, which matches statement a. Thus, this is true.

   b) Degree of the polynomial: The highest power of $x$ is $5$ (from $x^5$), so the degree of the polynomial is $5$. Statement b is also true.

   c) Leading coefficient: The leading term is the one with the highest degree, which is $x^5$. The coefficient of this term is $1$ (not $-7$), so statement c is false.

Conclusion:

The false statement is c) the leading coefficient of $p(x)$ is -7.

Would you like further details on any step?

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Here are 5 related questions:

1. How do you identify the degree of a polynomial?
2. What is a leading coefficient in a polynomial?
3. How do constant terms impact the overall behavior of a polynomial?
4. How would the graph of $p(x) = x^5 - 3x^4 + 2x^3 - 117649$ look?
5. Can the degree and leading coefficient change after simplifying a polynomial?

Tip: When analyzing polynomials, always arrange terms in descending order of degree for clarity.