1. **Write the polynomial in standard form.** The standard form orders terms from highest degree to lowest: $$f(x) = 3x^5 + x^3 + 2x + 4$$ 2. **Determine what $n$ should be for $f(x) = x^n$ to be a polynomial.** For $f(x)$ to be a polynomial, $n$ must be a non-negative integer. 3. **Identify the polynomial of degree 3.** A polynomial's degree is the highest exponent: - a. degree 2 - b. degree 3 (correct) - c. degree 4 - d. degree 5 4. **Find the polynomial with root at $x=3$.** This means $f(3) = 0$ or $(x-3)$ is a factor. Check which factor equals zero at 3: - a. $(3+3)(3+4) = 6*7 \neq 0$ - b. $3^2 - 6*3 + 9 = 9 - 18 + 9 = 0$ (correct) - c. $3+3=6 \neq 0$ - d. $3^2 + 6*3 + 9 = 9 +18 +9 = 36 \neq 0$ 5. **About $f(x) = 3x^4 - 2x^3 + x^2 - 5x + 7$:** The coefficient of $x^3$ is indeed $-2$. 6. **Factorize $2x^2 + 7x + 3$:** Factors of $2x^2 + 7x + 3$ are $(2x+1)(x+3)$. 7. **Find x-intercepts of $f(x) = (x - 2)(x + 3)^2(x - 4)(x + 1)$:** Roots are $x=2$, $x=-3$, $x=4$, and $x=-1$. 8. **Find y-intercept of $f(x)$:** Set $x=0$, $$f(0) = (-2)(3)^2(-4)(1) = (-2)(9)(-4)(1) = 72$$ 9. **Graph of polynomial with odd degree:** The graph rises on one end and falls on other. 10. **Polynomial from graph with roots at $-3, 1, 4$, rising right end, falling left end:** Leading term positive cubic polynomial likely $y = (x + 3)(x - 1)(x - 4)$ but choices differ - the closest is $y = (x + 1)(x - 4)(x - 3)$ but roots differ. The correct is a polynomial with these roots and positive leading coefficient. 11. **Select $a, n$ for $y = ax^n$ graph with negative leading coefficient, odd $n=3$: $a = -2$, $n=3$ matches graph descending left to right. 12. **Evaluate population at $t=2$ for $P(t) = 6t^4 - 5t^3 + 200t + 120000$:** $$P(2) = 6*(2)^4 - 5*(2)^3 + 200*2 + 120000 = 6*16 - 5*8 + 400 + 120000 = 96 - 40 + 400 + 120000 = 120456$$ 13. **Remainder when dividing $4x^3 - 3x^2 + 2x - 4$ by $x - 2$:** Use remainder theorem, evaluate at $x=2$: $$4(2)^3 - 3(2)^2 + 2(2) - 4 = 4*8 - 3*4 + 4 - 4 = 32 - 12 + 4 - 4 = 20$$ 14. **If $x=2$ is a root of $f(x) = x^3 - 4x^2 + 3x - 6$, then $f(2) = 0$ must be true. 15. **Factor of $f(x) = x^2 - 5x + 6$:** Factorization is $(x - 2)(x - 3)$ so factors include $(x - 3)$. 16. **True statement about $P(r)$:** If $P(r) = 0$, then $(x - r)$ is a factor of $P(x)$. 17. **Factored form of $x^2 + 5x + 6$ is $(x + 2)(x + 3)$. 18. **Process of rewriting polynomial as product of simpler polynomials is called factoring. 19. **Other factor of $x^2 - 9 = 0$ given $(x - 3)$ is $(x + 3)$. 20. **Polynomial equation with roots 1, -2, -4 in standard form:** $$(x - 1)(x + 2)(x + 4) = 0 o x^3 + 5x^2 + 2x - 8 = 0$$ 21. **Polynomial equation list:** i is polynomial, ii has negative exponents not polynomial, iii is polynomial equation. 22. **Roots of polynomial are values of $x$ satisfying equation; true. 23. **Polynomial with roots 3 and -4:** equals $(x - 3)(x + 4) = 0$. 24. **Roots of $x^2 + 7x + 10$: -5 and -2. 25. **Values satisfying $x^2 + 6x + 5 = 0$ are roots of factors $(x+1)(x+5)$ so roots are -1 and -5 - select correct root -1. 26. **Evaluate $P(-2)$ for $P(x) = 2x^3 - 5x + 7$:** $$2(-2)^3 - 5(-2) + 7 = 2(-8) + 10 + 7 = -16 + 10 + 7 = 1$$ 27. **Length of garden equals $rac{3x^2 + 5x - 2}{x + 2} = 3x - 1$ 28. **Polynomial that is polynomial function:= with non-negative integer exponents is $f(x) = 2x^2 - rac{3}{2}x^2$ 29. **Leading coefficient of $f(x) = 2x + x^3 + 4$ is 1 coefficient of highest degree term $x^3$. Answers summary: 19.d, 20.b, 21.b, 22.b, 23.c, 24.a, 25.a, 26.a,27.c,28.a, 29.d,30.b,1.d,2.a,3.c,4.a,5.b,6.a,7.a,8.d,9.c,11.a,12.c,13.b,14.b,15.a,16.c,17.b,18.a.