given the polynomial p(x) = -3x^4 + 2x^3 -7^6 +x^5, which of the following is FALSE? a) the constant term of p(x) is -117649 b) the degree of p(x) is 5 c) the leading coefficient of p(x) is -7

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.

Given the polynomial p(x) = -3x^4 + 2x^3 - 7^6 + x^5, which of the following is FALSE? a) the constant term of p(x) is -117649 b) the degree of p(x) is 5 c) the leading coefficient of p(x) is -7

Learn how to analyze the polynomial p(x) = x^5 - 3x^4 + 2x^3 - 117649. This problem covers the degree, leading coefficient, and constant term of polynomials, with detailed explanations of each. Suitable for students in Grades 9-12.

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