What is the standard form of any quadratic function?
ax^2 + bx + c
100
What name is given to the "shortcut" process of polynomial division that is useful for finding remainders and roots?
Synthetic division
100
Given any arbitrary polynomial, which terms are the most important to consider when applying the Rational Root Theorem?
Degree term and constant term (first and last)
100
Which principle states that given a polynomial function f(x) with real coefficients, given two numbers a and b where f(a) and f(b) are differing in sign, that a zero to f(x) must exist between a and b?
Location Principle
200
What word is used to identify the highest power of a variable present in a given polynomial?
Degree
200
Suppose a quadratic function has its zeroes at x=3 and x=-2. What is the quadratic function with these zeroes?
x^2 - x + 6
200
Given a polynomial function with f(3) = 25, what would be the remainder after dividing the given polynomial by the term (x-3)?
25
200
Give a list of the possible rational roots of the polynomial
3x^2 - 4x + 2.
Possible rational roots:
+1, -1
+1/3, -1/3
+2/3, -2/3
200
Determine between which consecutive integers the real zeroes of
f(x) = 12x^3 - 20x^2 - x + 6
are located.
Between -1 and 0, 0 and 1, 1 and 2
300
Which "really important" theorem states that any polynomial function of degree n will always have n number of complex roots?
The Fundamental Theorem of Algebra
300
Solve the following quadratic equation by factoring:
f(x) = 4x^2 + 16x - 84
x = 7; x = -3
300
Use the Remainder Theorem to find the remainder when 2x^3-3x^2+x is divided by x-1, and use that information to state whether x-1 is a factor of the polynomial.
Remainder is 0; thus, x-1 is a factor
300
Use Descartes' Rule of Signs to identify the number of potential positive and negative real zeroes of the polynomial
x^3 - 4x^2 - 8x + 12
Possible positive roots: 2 or 0
Possible negative roots: 1
300
Approximate the real zeroes of
f(x) = -3x^4 + 16 x^3 - 18x^2 + 5
to the nearest tenth.
x = -0.4, 1.0, 3.8
400
Suppose a given polynomial function is of the form
f(x) = (x-4)(x^2-9)(x^2+16).
Find the roots of f(x).
x = 4; x = -3; x = 3; x = -4i; x = 4i
400
Solve the following equation below using the Quadratic Formula:
x^2 - 5x + 9 = 0
x = (5 + i*sqrt(11))/2; x = (5 - i*sqrt(11))/2
400
Given the polynomial from the previous question,
2x^3-3x^2+x,
find the depressed polynomial after division by x - 1, and use that to find all zeroes of the polynomial.
Depressed polynomial: 2x^2 -x
Factors: x = 0, 1/2, 1
400
Find the number of possible positive and negative roots of the polynomial
x^3 + 8x^2 +16x + 5,
and then state those roots.
Possible positive roots: none
Possible negative roots: 3 or 1; x = -5, x = (-3 + sqrt(5))/2, x = (-3 - sqrt(5))/2
400
Use the Upper Bound Theorem to find an integral upper bound and the Lower Bound Theorem to find an integral lower bound for the zeroes of the polynomial
f(x) = 6x^3 - 7x^2 - 14x + 15
Lower bound: -2
Upper bound: 3
500
Using the polynomial function from before,
f(x) = (x-4)(x^2-9)(x^2+16)
find the standard form of hte polynomial.
x^7-4x^6+27x^4+16x^3-64x^2-144x+576
500
Solve the following equation by completing the square:
3x^2 - 12x = -4
x = 2 + (2*sqrt(6))/3; x = 2 - (2*sqrt(6))/3
500
Determine the binomial factors and respective zeroes of the polynomial
x^3 - 7x + 6.
Factors: (x+3)(x-1)(x-2)
Zeroes: x = -3, 1, 2
500
Find the number of possible positive and negative real zeroes for f(x) = 2x^5+3x^4-6x^3+6x^2-8x+3. Then determine the rational zeroes of the polynomial.
Possible positive zeroes: 4, 2, or none
Possible negative zeroes: 1
Rational zeroes: -3, 1, 1/2
500
The Toaster Treats company uses a box with a square bottom to package its product. The height of the box is 3 inches more than the length of the bottom. Find the dimension of the box given that the volume of materials is 42 cubic inches.