1. The problem asks to determine if the statements about polynomial functions and zeros are true or false. 2. Statement 1: "Every quadratic function is a polynomial function." This is true because a quadratic function is a polynomial of degree 2. 3. Statement 2: "If a real number c is a zero of a polynomial function f, then x - c is not a factor of f." This is false because if c is a zero, then by the Factor Theorem, x - c is a factor. 4. Statement 3: "f(x) = 0 is polynomial function with no degree assigned to it." This is true; the zero polynomial is defined to have no degree. 5. Statement 4: "Let f be a polynomial function such that f(4) = 2 and f(7) = -1. Then there is at least one zero of f between 4 and 7." This is true by the Intermediate Value Theorem since the function changes sign. 6. For question 5: Given x - 2 is a factor of f(x) = kx^2 + 3x - 18, then f(2) = 0. Calculate: $$k(2)^2 + 3(2) - 18 = 0 \Rightarrow 4k + 6 - 18 = 0 \Rightarrow 4k - 12 = 0 \Rightarrow 4k = 12 \Rightarrow k = 3$$ Answer: D. 3 7. Question 6: Leading coefficient of g(x) = 7x - 2x^8 + 9x^3 - 5 is the coefficient of the highest degree term. Highest degree term is $-2x^8$, so leading coefficient is -2. Answer: D. -2 8. Question 7: Factor of h(x) = x^3 + 1. Use sum of cubes factorization: $$x^3 + 1 = (x + 1)(x^2 - x + 1)$$ So factor is x + 1. Answer: B. x + 1 9. Question 8: Which is polynomial? A: $3x^{13} + \frac{7}{x^2}$ has negative exponent, not polynomial. B: $2x^3 + 5|x|$ has absolute value, not polynomial. C: $x^{23} - 8x^4 + \sqrt{2}$ is polynomial. D: $x^{-6} - 4x + 9$ has negative exponent, not polynomial. Answer: C 10. Question 9: 3 is zero of $f(x) = x^3 - 2x^2 + tx + 3$. So $f(3) = 0$: $$27 - 18 + 3t + 3 = 0 \Rightarrow 12 + 3t = 0 \Rightarrow 3t = -12 \Rightarrow t = -4$$ Answer: C. -4 11. Question 10: 6 is zero of f. Which is NOT true? A: $f(6) > 0$ is not necessarily true since zero means $f(6) = 0$. B, C, D are true by definition of zero and factor. Answer: A 12. Question 11: When $x^3 + ax^2 + x - 5$ divided by $x - 2$, remainder is -3. By Remainder Theorem: $$f(2) = 2^3 + a(2)^2 + 2 - 5 = -3$$ $$8 + 4a + 2 - 5 = -3 \Rightarrow 5 + 4a = -3 \Rightarrow 4a = -8 \Rightarrow a = -2$$ Answer: B. -2 13. Question 12: Degree of f is 7, degree of g is 4. Degree of fg is sum: 7 + 4 = 11. Degree of f + g is max degree: 7. Degree of f - g is max degree: 7. Degree of f/g is not defined as product of degrees. Answer: A 14. Question 13: Zeros of $f(x) = x^3 - 4x$. Factor: $$x(x^2 - 4) = x(x - 2)(x + 2)$$ Zeros: 0, 2, -2. Answer: A 15. Question 14: Find polynomial degree 3 with zeros -5, 2, 4 and $f(3) = -24$. Form: $$f(x) = k(x + 5)(x - 2)(x - 4)$$ Find k using $f(3) = -24$: $$k(3 + 5)(3 - 2)(3 - 4) = -24 \Rightarrow k(8)(1)(-1) = -24 \Rightarrow -8k = -24 \Rightarrow k = 3$$ So: $$f(x) = 3(x + 5)(x - 2)(x - 4)$$ 16. Question 15: Remainder when $f(x) = x^{33} + 4$ divided by $x - 1$. By Remainder Theorem: $$f(1) = 1^{33} + 4 = 1 + 4 = 5$$ Final answers: 1. True 2. False 3. True 4. True 5. D 6. D 7. B 8. C 9. C 10. A 11. B 12. A 13. A 14. $f(x) = 3(x + 5)(x - 2)(x - 4)$ 15. 5