The Basics
Multiplying Polynomials
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

What property do we use to multiply polynomials?

We use the Distributive Property to multiply polynomials.

1/(x-3)^2

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
List the first 6 rows of Pascal's Triangle
We will have to write this answer out on the board ...
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
Expand the following binomial: (y + 3)^3
(y + 3)^3 = y^3 + 9y^2 + 27y + 27
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
Multiply the following polynomials: (a - 3) (2 - 5a + a^2)
(a - 3) (2 - 5a + a^2) = a^3 - 8a^2 + 17a - 6
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
Whiteboard Daily Double!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Wager the amount of money you would like to gamble on this problem. Mr. Etz will write down the question on the board. Good Luck!
The answer is ...
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
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