The Rational Root Theorem and Polynomial Equations
🔵🟡 Rational Root Quest 🟡🔵
Welcome, Math Champion! 🌟 This quiz will challenge your skills in The Rational Root Theorem and Polynomial Equations. Test possible roots, factor carefully, and learn from every solution! 🚀
Part I — Fundamentals 🧠
Choose the best answer. These questions test important ideas before solving.
1. For 5x³ − 4x² + 7x − 12 = 0, which list gives the possible values of p?
2. For 8x⁴ − 3x² + 10x − 15 = 0, possible values of q are:
3. A rational root is a root that can be written as:
4. How do we know that a possible rational root is an actual root?
5. If −3/4 is a root of a polynomial equation, which factor corresponds to it?
6. If 5/2 is a root, which factor corresponds to it?
7. Which statement about 0 as a root is true?
8. A quartic polynomial equation can have at most how many real roots?
9. If a polynomial has leading coefficient 1, its rational root candidates are:
10. What does it mean if a root is repeated?
Part II — Solving Questions ✏️
Solve each polynomial equation. Use rational root testing, grouping, and factoring.
11. Solve x³ − 9x² + 23x − 15 = 0.
12. Solve 2x³ − 7x² + 2x + 3 = 0.
13. Solve x³ − 2x² − 9x + 18 = 0.
14. Solve x⁴ − 13x² + 36 = 0.
15. Solve 3x³ − 2x² − 19x − 6 = 0.
16. Find k if x + 3 is a factor of f(x) = x³ + kx² − 4x − 6.
17. Find k if the remainder is 10 when f(x) = x³ − 2x² + kx + 4 is divided by x − 3.
18. Solve x³ − 6x² + 3x + 10 = 0.
19. Solve 2x⁴ + 3x³ − 8x² − 12x = 0.
20. Solve 6x³ − 5x² − 17x + 6 = 0.
Part III — Dropdown Challenge 🎯
Each question has a dropdown with the 10 choices. Choose carefully and check your work! 💙💛
Answer Choices:
A. −5
B. −3
C. −1
D. 0
E. 1
F. 2
G. 3
H. 5
I. {−4, 1, 2}
J. {−1, 1/2, 4}
A. −5
B. −3
C. −1
D. 0
E. 1
F. 2
G. 3
H. 5
I. {−4, 1, 2}
J. {−1, 1/2, 4}
21. Find k if x − 1 is a factor of x³ + kx² + 3x + 1.
22. The polynomial x³ + 2x² − 5x − 6 = 0 has roots 2, −1, and one more rational root. What is the missing root?
23. Find one negative rational root of x³ + x² − 4x − 4 = 0.
24. Find the smallest nonnegative rational root of x⁴ − 9x² = 0.
25. Find one positive rational root of x³ − 2x² − x + 2 = 0.
26. Find k if x + 1 is a factor of x³ + 4x² + kx − 1.
27. Find one rational root of x³ − 6x² + 11x − 6 = 0.
28. Find the missing root of x³ − 8x² + 7x + 30 = 0 if two roots are −2 and 3.
29. Solve x³ + x² − 10x + 8 = 0.
30. Solve 2x³ − 7x² + 5x + 4 = 0.