Polynomial Vocabulary
What is the degree of the polynomial 4x^3 - 2x + 7?
3rd
(x + 3) + (2x - 1)
3x + 2
Factor: x^2 - 9
(x - 3)(x + 3)
What is the Zero Product Property?
If ab = 0, then a = 0 or b = 0
Easy: If f(x) = x^2 + 3x + 2, find f(2)
2^2 + 3*2 + 2 = 4 + 6 + 2 = 12
Define a monomial and give an example.
A monomial is a polynomial with only one term. Example: 7x^2
Multiply: (x+2)(x−5)
x^2 - 3x - 10
: Factor completely: x^2 + 5x + 6
(x + 2)(x + 3)
State the Remainder Theorem.
If a polynomial f(x) is divided by x - c, the remainder is f(c)
Easy+1:Let f(x) = x^2 - 4x + 3. Find the sum of the roots.
4
What is the leading coefficient of -5x^4 + 3x^2 - 8?
-5
Divide x^3 + 2x^2 - x - 2 by x + 1 using long division.
Quotient: x^2 + x - 2, Remainder: 0
Factor by grouping: x^3 + 3x^2 + 2x + 6
(x^2 + 2)(x + 3)
Use the Rational Root Theorem to list all possible rational roots of 2x^3 + 3x^2 - x - 6
±1, ±2, ±3, ±6, ±1/2, ±3/2
Medium: Find the polynomial of least degree with roots 2, -1, and 3.
(x - 2)(x + 1)(x - 3) = x^3 - 4x^2 - x + 6
What makes a polynomial "quadratic"?
It's a polynomial of degree 2.
Use synthetic division to divide 2x^3 - 3x^2 + x + 6 by x - 2.
2x^2 + x + 3, Remainder: 12
Factor x^4 - 16
(x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4)
Given that x = 2 is a root of x^3 - 3x^2 - 4x + 12, factor the polynomial completely.
(x - 2)(x^2 - x - 6) = (x - 2)(x - 3)(x + 2)
Medium+1: If f(x) = x^3 + ax^2 + bx + c and f(1) = 0, what must be true about the coefficients?
1 + a + b + c = 0
Define a homogeneous polynomial and give a non-trivial example in two variables.
All terms have the same total degree. Example: x^3 + 3x^2y + 3xy^2 + y^3
If f(x) = x^4 + 2x^3 - x + 3, find the remainder when divided by x^2 + 1.
-3x+4
Factor x^5 - x completely over the integers.
x(x^4 - 1) = x(x^2 - 1)(x^2 + 1) = x(x - 1)(x + 1)(x^2 + 1)
Let P(x) = x^3 - 6x^2 + 11x - 6. Without factoring, show that all roots are real and distinct.
Try rational roots: x = 1, x = 2, and x = 3 all work. So roots are real and distinct.
SUPER DUPER LUPER WOOPER GUUPER DECENTLY hard: Let P(x) be a polynomial of degree 4 such that P(1) = P(2) = P(3) = P(4) = 0, and P(5) = 120. Find P(x).
P(x) = a(x - 1)(x - 2)(x - 3)(x - 4)
Plug in x = 5:
120 = a(4)(3)(2)(1) = 24a → a = 5
So,
Answer: P(x) = 5(x - 1)(x - 2)(x - 3)(x - 4)