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The image you uploaded contains the following math-related content regarding polynomial functions:

1. **General Characteristics of Polynomial Functions**:
   - The question asks to discuss the common characteristics of polynomial functions, including degree, concavity, zeros, and turning points.

2. **Newton’s Divisor Algorithm, Remainder Theorem, Vieta's Rules, and Rational Root Test**:
   - The second part of the problem is focused on various algorithms and theorems related to polynomials, such as Newton's Divisor Algorithm, Remainder Theorem, Vieta’s Formulas, and Rational Root Theorem.
   - You are asked to apply these theorems to solve specific polynomial problems:
     1. **Derivative of a polynomial**: Find the first derivative of \( f(x) = x^4 - 5x^3 + 2x^2 - 7x + 3 \).
     2. **Zeros of a polynomial**: Find the zeros (roots) of the polynomial \( f(x) = 6x^3 + 5x^2 - 6x - 1 \).

Would you like a detailed explanation and solution for each part of these polynomial problems? 

Here are five relative questions to expand your understanding:
1. What are the general forms and characteristics of polynomial functions?
2. How does Vieta's Rule help in solving for the roots of polynomials?
3. What is Newton’s Divisor Algorithm, and how is it applied to polynomials?
4. How is the Rational Root Theorem used to determine possible rational roots of polynomials?
5. How does the Remainder Theorem help in simplifying polynomial division?

**Tip**: Understanding the degree of a polynomial is key to predicting its general shape and number of roots.

Discuss the common characteristics of polynomial functions. Use Newton’s Divisor Algorithm, Remainder Theorem, Vieta’s Rule, and Rational Root Test to solve the given polynomial problems: (1) Find the first derivative of f(x) = x^4 - 5x^3 + 2x^2 - 7x + 3, (2) Find the zeros of f(x) = 6x^3 + 5x^2 - 6x - 1.

Explore the characteristics of polynomial functions and learn how to apply Newton’s Divisor Algorithm, Remainder Theorem, Vieta’s Rule, and Rational Root Theorem. Includes a detailed solution to finding the derivative of f(x) = x^4 - 5x^3 + 2x^2 - 7x + 3 and the zeros of f(x) = 6x^3 + 5x^2 - 6x - 1.

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