The Basics
Dividing Polynomials
Factoring Polynomials
Finding those Roots
100
What is a polynomial?
A polynomial is a monomial or a sum or difference of monomials.
100
If the remainder after synthetic division is NOT zero, is (x - a) a factor of P(x)?
(x - a) is only a factor of P(x) if the remainder after synthetic division is zero. So if the remainder is NOT zero, (x - a) is NOT a factor.
100
Factor the following polynomial: x^2 - 17x - 38
x^2 - 17x - 38 = (x + 2) (x - 19)
100
What is the difference between a "zero" of a polynomial and a "root" of a polynomial?
There is no difference!
200
What is the degree of the following polynomial: 3x^(2) - 4 +8x^(4)
The degree of 3x^(2) - 4 +8x^(4) is 4.
200
If (x + 1/2) is our divisor, what do we put in the little box in order to perform synthetic division?
If (x + 1/2) is our divisor, we set (x - 1/2) equal to 0 to find what we put in the little box - in this case, it would be 1/2.
200
Factor the following polynomial: 4d^4 + 108d
4d^4 + 108d = 4d (d^3 + 27)
200
You can earn bonus points with this question: Write down all statements that are equivalent to the following sentence: "a is a root of the polynomial P(x)"
Possible answers: P(a) = 0 a is an x-intercept of the graph of P(x) (x - a) is a factor of P(x) When you divide P(x) by (x - a), the remainder is 0 a is a zero of P(x)
300
Subtract the following polynomials: (34 + 8x^3 - 9x^2) - (3x^3 + 10x^2 - 4x - 4)
(34 + 8x^3 - 9x^2) - (3x^3 + 10x^2 - 4x - 4) = 5x^3 - 19x^2 + 4x + 38
300
Divide the following polynomials: (-y^2 + 2y^3 + 25) / (y - 3)
(-y^2 + 2y^3 + 25) / (y - 3) = 2y^2 + 5y + 15 + (70/y-3)
300
Factor the following polynomial by grouping: 4x^3 - 8x^2 + 9x - 18
4x^3 - 8x^2 + 9x - 18 = (x - 2) (4x^2 + 9)
300
Is (x + 2) a factor of P(x) = -2x^3 + 7x^2 - 3x - 9 ?
(x + 2) is NOT a factor of P(x) = -2x^3 + 7x^2 - 3x - 9, because the remainder after synthetic division is NOT zero.
400
If h(x) = x^2 - 6x + 3, find h(-2)
If h(x) = x^2 - 6x + 3, h(-2) = (-2)^2 - 6(-2) + 3 = 19
400
Is (x - 4) a factor of x^3 - 4x^2 + 3x - 5 ?
Since x^3 - 4x^2 + 3x - 5 / (x - 4) does not have a remainder of 0, (x - 4) is not a factor of x^3 - 4x^2 + 3x - 5
400
TEST - You will receive 10 points for factoring the following polynomial by grouping: 4x^3 - 8x^2 + 9x - 18
Remember to pay attention!
400
Find the roots of f(x) = x^2 + 3x - 5
The roots of f(x) = x^2 + 3x - 5 are x = (Mr. Etz will write the answer on the board)
500
Use a graphing calculator to find the zeros of f(x) = 2x^2 + 5x + 3
The zeros of f(x) = 2x^2 + 5x + 3 are -1 and -3/2
500
Write the simplest polynomial with roots at x = -2, 3, and 1. Write your answer in standard form.
The simplest polynomial with roots at x = -2, 3, and 1 is x^3 - 2x^2 - 5x + 6
500
Whiteboard Daily Double!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Mr. Etz will write down the question on the white board - make your wagers now.
And the answer is ...
500
This is the most difficult question in our review game, and it will be worth 1000 points: Find all the roots of P(x) = x^4 - x^3 + 7x^2 - 9x - 18
The roots of P(x) = x^4 - x^3 + 7x^2 - 9x - 18 are: -1, 2, 3i, and -3i
M
e
n
u