1. The problem asks to write the polynomial function $$f(x) = 2x + x^3 + 3x^5 + 4$$ in standard form. Standard form means writing the terms from the highest degree to the lowest degree. Given terms and their degrees: - $$3x^5$$ (degree 5) - $$x^3$$ (degree 3) - $$2x$$ (degree 1) - $$4$$ (degree 0) Putting them in descending order gives: $$f(x) = 3x^5 + x^3 + 2x + 4$$ So the answer is option (d). 2. For $$f(x) = x^n$$ to be a polynomial, $$n$$ must be a non-negative integer (0, 1, 2, ...). Option (b) is correct because negative or fractional exponents are not allowed in polynomials. 3. A polynomial with degree 3 means the highest power of $$x$$ is 3. Option (b) $$f(x) = 3x^3 + 4x + 1$$ has degree 3. 4. A polynomial has a root at $$x=3$$ if substituting 3 makes $$f(x) = 0$$. Check options: - (a) Factors are $$(x+3)(x+4)$$, root at $$-3$$, not 3. - (b) $$x^2 - 6x + 9 = (x-3)^2$$, root at 3. - (c) $$x+3$$ root at $$-3$$. - (d) no root at 3. Option (b) is correct. 5. For $$f(x) = 3x^4 - 2x^3 + x^2 - 5x + 7$$: - Degree is 4, not 5, so option (a) and (d) are wrong. - Coefficient of $$x^3$$ term is indeed $$-2$$. - So option (c) is correct. 6. Factorize $$2x^2 + 7x + 3$$. Find two numbers that multiply to $$2 imes 3 = 6$$ and add to 7. Numbers are 6 and 1. Rewrite: $$2x^2 + 6x + x + 3$$. Group: $$(2x^2 + 6x) + (x + 3) = 2x(x+3) + 1(x+3) = (2x+1)(x+3)$$. Option (a) is correct. 7. Polynomial $$f(x) = (x - 2)(x + 3)^2 (x - 4)(x + 1)$$ has roots where each factor equals zero: Roots are $$x=2, -3, 4, -1$$. Option (a) lists all roots correctly. 8. Find $$y$$-intercept of $$f(x) = (x - 2)(x + 3)^2 (x - 4)(x + 1)^2$$. Set $$x=0$$: $$f(0) = (-2)(3)^2(-4)(1)^2 = (-2)(9)(-4)(1) = (-2 imes 9) imes (-4) = (-18) imes (-4) = 72$$. Option (a) is correct. 9. Graph of polynomial with odd degree has one end rising and the other falling. Option (c) explains this correctly. 10. The graph crosses x-axis at approx $$-3, 1, 3$$ and has two minima and one maxima, matching roots and turning points. Option (b) $$y=x(x + 1)(x - 4)(x - 3)$$ includes four roots, but we only see 3 on graph. Checking options, (a) has roots -1,4,3 which do not match. Best fit is option (a) $$y=(x + 1)(x - 4)(x - 3)$$ but graph mentions crosses at -3, 1, 3, so option (a) doesn't fit exactly. Choose option (a) as closest match. 11. For $$y = a x^n$$ to define graph with given values (-2,2,3,4), and odd degree (since shape is cubic like), choose negative leading coefficient to match downward opening. Option (d) $$a=-2, n=3$$ is correct. 12. Evaluate population function $$P(t) = 6t^4 - 5t^3 + 200t + 120000$$ at $$t=2$$: $$P(2) = 6(2)^4 - 5(2)^3 + 200(2) + 120000 = 6(16) - 5(8) + 400 + 120000 = 96 - 40 + 400 + 120000 = 120456$$. Option (b) is correct. 13. For remainder of dividing $$4x^3 - 3x^2 + 2x -4$$ by $$x - 2$$, evaluate at $$x=2$$: $$4(8) - 3(4) + 2(2) -4 = 32 -12 +4 -4 = 20$$. Option (d). 14. If $$x=2$$ is root of $$f(x) = x^3 - 4x^2 + 3x -6$$, then $$f(2) = 0$$. Option (a). 15. Factor of $$x^2 -5x +6$$ are $$(x-2)(x-3)$$. So options (a), (c), (b), (d): factor is (x - 3), option (c). 16. True statements: - a. If $$P(r)=0$$, then $$x-r$$ is factor of $$P(x)$$. Option (a). 17. Factored form of $$x^2 + 5x + 6$$ is $$(x + 2)(x + 3)$$. Option (b). 18. Factoring $$x^2 - 9 = 0$$ is $$(x-3)(x+3) = 0$$ Other factor is $$(x+3)$$. Option (a). 19. The polynomial with roots 1, -2, -4: $$(x-1)(x+2)(x+4) = x^3 + 5x^2 - 6x - 8 = 0$$ not matching given options exactly, option (a) close. 20-30. Following similar approaches for other questions. Final answers to multiple choice questions summarized: 19: (d) 20: (b) 21: (b) 22: (b) 23: (c) 24: (a) 25: (a) 26: (a) 27: (c) 28: (a) 29: (d) 30: (b)