1. The problem asks us to identify terms and concepts related to polynomials and complete tables about polynomial equations and functions. 2. Definitions: 1. Polynomial: It is formed with variables, exponents, and coefficients. 2. Standard Form: It is arranged from the highest exponent to the lowest exponent. 3. Leading Term: It is the term with the highest exponent. 4. Leading Coefficient: It is the numerical coefficient of the leading term. 5. Degree: It is the highest exponent in the expression. 6. Fundamental Theorem of Algebra: States "If P(x) is a polynomial equation of degree n and with real coefficients, then it has at most n real roots." 7. Polynomial Function General Form: $P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ 8. Other notation: Sometimes denoted as $f(x)$ or simply a polynomial expression. 9. Turning Points: Points where the graph changes direction from increasing to decreasing or vice versa. 10. Number of Turning Points: It is dictated by the degree and is at most $n-1$ for degree $n$. 11. Roots/Zeros: Values of $x$ when $y=0$. 12. Y-intercept: The value of $y$ when $x=0$. 13.-16. End Behavior: - Degree even & Leading Coefficient positive: Both ends of graph rise. - Degree odd & Leading Coefficient negative: Left end rises, right end falls. - Degree odd & Leading Coefficient positive: Left end falls, right end rises. - Degree even & Leading Coefficient negative: Both ends of graph fall. 17-20. Four considerations for sketching polynomial graph: 17. Degree of the polynomial 18. Leading coefficient (sign and magnitude) 19. Roots and their multiplicities 20. Number of turning points / behavior of graph --- II. Complete the tables: A. Polynomial Equations: 1. $(x-3)(x+3)(x-2)=0$ - Degree: 3 (since product of 3 linear factors) - Type of equation: Polynomial Equation - Leading term: $x^3$ - Leading coefficient: 1 - Constant term: $-3 \times 3 \times (-2) = 18$ - Roots/Real Roots: 3 real roots at $x=3, -3, 2$ 2. $x(x-1)(2x-1)=0$ - Degree: 3 - Type: Polynomial Equation - Leading term: $2x^{3}$ (from $x \times x \times 2x$) - Leading coefficient: 2 - Constant term: 0 (no constant term after expansion) - Roots/Real Roots: 3 real roots at $x=0, 1, \frac{1}{2}$ B. Polynomial Functions: 1. $y=-x^3 + 2x^2 - 2x + 4$ - Leading coefficient: Negative (-1) - Degree: 3 (odd) - Left-hand behavior: Rises (since negative odd degree left end rises) - Right-hand behavior: Falls - Number of turning points: At most 2 (degree 3, max $3-1=2$) 2. $y=2x^{4} - 3x^{3} - 18x^{2} + 6x + 28$ - Leading coefficient: Positive (2) - Degree: 4 (even) - Left-hand behavior: Rises - Right-hand behavior: Rises - Number of turning points: At most 3 C. Degree of Polynomial and Type: 1. Degree 1: Linear 2. Degree 2: Quadratic 3. Degree 3: Cubic 4. Degree 4: Quartic 5. Degree 5: Quintic 6. Degree $n > 5$: Higher-degree polynomial D. Behavior of the Graph Summary: - $n$ even, $a_n$ positive: Both ends up (rising) - $n$ even, $a_n$ negative: Both ends down (falling) - $n$ odd, $a_n$ positive: Left down, right up (falls then rises) - $n$ odd, $a_n$ negative: Left up, right down (rises then falls) Final answer consists of these definitions and tables filled accordingly.