Remainder Theorem, Factor Theorem, and Factoring Polynomials
✨ Polynomial Power Quiz ✨
Test your knowledge about the Remainder Theorem, Factor Theorem, and Factoring Polynomials. Read carefully, solve confidently, and have fun! 🌟
Part I — Fundamentals 🧠
Choose the best answer. These questions test your understanding of the main concepts.
1. The Remainder Theorem states that when a polynomial f(x) is divided by x − a, the remainder is:
2. According to the Factor Theorem, x − a is a factor of f(x) if and only if:
3. If x + 4 is a factor of f(x), then which value of x should make f(x) = 0?
4. If a polynomial has zeros 2, −3, and 5, then one possible factored form is:
5. The remainder when f(x) is divided by x + 7 can be found by evaluating:
6. If f(3) = 8, then the remainder when f(x) is divided by x − 3 is:
7. If f(−2) = 0, then which binomial is a factor of f(x)?
8. Which expression represents a polynomial of degree 4 with zeros −1, 2, 3, and 6?
9. What does it mean if the remainder is 0 when f(x) is divided by x − a?
10. Which of the following is the correct factored form of x² − 9?
Part II — Solving Questions ✏️
Solve each polynomial problem. These are still multiple choice, but you need to calculate carefully.
11. Find the remainder when f(x) = 2x³ − 5x² + 4x − 7 is divided by x − 2.
12. Find the remainder when g(x) = x⁴ − 3x³ + 2x² − 5 is divided by x + 1.
13. Determine whether x − 3 is a factor of f(x) = x³ − 6x² + 11x − 6.
14. Determine whether x + 2 is a factor of h(x) = 2x³ + 5x² − 4x − 12.
15. Find f(−1) if f(x) = 3x⁴ − 2x³ + x − 9.
16. Find the remainder when p(x) = 4x³ + x² − 6x + 8 is divided by x + 2.
17. Which of the following is the complete factorization of x³ − 4x² − x + 4?
18. A polynomial of degree 3 has zeros −2, 1, and 4. Which is a possible polynomial?
19. Find the value of k if x − 1 is a factor of f(x) = x³ + kx² − 4x + 2.
20. Find the value of k if the remainder is 5 when f(x) = 2x³ + kx² − 3x + 1 is divided by x − 2.
Part III — Dropdown Challenge 🎯
Each question has a dropdown. Choose from the 10 choices below. Think carefully because some answers may look similar!
Answer Choices:
A. −30
B. −15
C. −6
D. 0
E. 3
F. 5
G. 12
H. 18
I. (x − 2)(x + 1)(x − 3)
J. (x + 4)(x − 1)(x + 2)
A. −30
B. −15
C. −6
D. 0
E. 3
F. 5
G. 12
H. 18
I. (x − 2)(x + 1)(x − 3)
J. (x + 4)(x − 1)(x + 2)
21. Find the remainder when f(x) = x³ − 4x² + x + 6 is divided by x − 3.
22. Find the remainder when g(x) = 2x³ + x² − 5x − 3 is divided by x + 2.
23. Determine the remainder when h(x) = x⁴ − 2x³ − 7x² + 8x + 12 is divided by x − 2.
24. Find p(−1) if p(x) = 3x³ + 2x² − 4x + 2.
25. Find the remainder when q(x) = x³ + 3x² − 10x − 24 is divided by x + 4.
26. If x − 2, x + 1, and x − 3 are factors of a cubic polynomial, which choice gives its factored form?
27. If x + 4, x − 1, and x + 2 are factors of a cubic polynomial, which choice gives its factored form?
28. Find f(1) if f(x) = 5x⁴ − 3x³ + 2x² − x.
29. Find the remainder when r(x) = x³ − 6x² + 12x − 8 is divided by x − 2.
30. Find the remainder when s(x) = 2x⁴ − 3x³ + x² − 4x + 2 is divided by x + 1.