FTA & Degree
According to the Fundamental Theorem of Algebra, how many total zeros (real and complex) does the polynomial function f(x) = x^4 - 3x^3 + 2x^2 - 7 have?
4
According to the Complex Conjugates Theorem, if 3 + i is a zero of a polynomial function with real coefficients, what other number must also be a zero?
3 - i
When using synthetic division to test if x = 3 is a zero of a polynomial, what must the final remainder be?
0 (zero)
Descartes' Rule of Signs uses the number of sign changes in f(x) to determine the possible number of which type of zeros?
Positive real zeros
What fruit is traditionally bobbed for at Halloween parties?
Apples
If a polynomial function has a degree of 5, what is the maximum number of REAL zeros it can have?
5
According to the Irrational Conjugates Theorem, if 2 + \sqrt{5} is a root of a polynomial equation with rational coefficients, what is another root?
2 - \sqrt{5}
What are all the zeros of the function f(x) = x(x - 4)(x^2 + 9) ?
x = 0, 4, 3i, -3i
How many sign changes occur in the coefficients of f(x) = 2x^4 - 5x^3 - x^2 + 3x - 7 ?
3 sign changes (Positive to Negative, Negative to Positive, Positive to Negative)
Which classic monster is vulnerable to a silver bullet?
Werewolf
How many total complex roots does the equation x^3 + 8 = 0 have?
3
What is the product of the complex conjugate pair (3 + i) and (3 - i) ?
10 since 3^2 - i^2 = 9 - (-1) = 10
Completely factor x^3 - 2x^2 - 9x + 18 by grouping.
(x - 2)(x - 3)(x + 3)
If f(x) has 3 sign changes, what are the possible numbers of positive real zeros?
3 or 1 (Subtracting multiples of 2)
What is the title of the famous 1818 novel by Mary Shelley that features a scientist creating a creature?
Frankenstein
True or False: A polynomial of degree "n" can have more than "n" zeros if some are imaginary.
False (It has exactly n zeros, counting multiplicity).
Write a quadratic polynomial function f(x) with a leading coefficient of 1 that has the zeros x = 2i and x = -2i.
f(x) = x^2 + 4
If synthetic division of p(x) by (x - 2) yields a quotient of x^2 + 3x + 2 with a remainder of 0, what is the complete factorization of p(x) into linear factors?
(x - 2)(x + 1)(x + 2)
For f(x) = x^3 + 4x^2 + 5x + 2 , how many positive real zeros can this function have?
0 (There are no sign changes in f(x))
In what country did the holiday of Halloween originate from the ancient Celtic festival of Samhain?
Ireland
If a polynomial p(x) has degree 6 and 4 of its roots are non-real complex numbers, how many real roots must it have?
2 real roots (since non-real complex roots come in conjugate pairs, 6 - 4 = 2 ).
What is the lowest possible degree of a polynomial function with rational coefficients that has zeros x = 1 , x = \sqrt{3} , and x = 2 - i ?
Degree 5 (Zeros must be 1, \sqrt{3} , -\sqrt{3} , 2 - i , and 2 + i ).
Find all zeros of the polynomial function f(x) = x^3 - 3x^2 + 4x - 12
x = 3, 2i, -2i
If a 4th-degree polynomial has 0 positive real zeros and 2 negative real zeros, how many imaginary zeros must it have?
2 imaginary zeros
What vegetable was originally carved into Jack-o'-lanterns before pumpkins became popular in America?
Turnips (or Potatoes/Beets)