The Basics
A ______ is a monomial or a sum or difference of monomials.
What is a polynomial?
What property do we use to multiply polynomials?
We use the Distributive Property to multiply polynomials.
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.
Factor the following polynomial: x2 - 17x - 38
x2 - 17x - 38 = (x + 2) (x - 19)
What is the difference between a "zero" of a polynomial and a "root" of a polynomial?
There is no difference!
What is the degree of the following polynomial: 3x2 - 4 +8x4
The degree of 3x2 - 4 +8x4 is 4.
Write the equation for the difference of cubes.
The difference of cubes is: (a - b)(a2 + ab + b22
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.
Factor the following polynomial: 4d4 + 108d
4d4 + 108d = 4d (d3 + 27) = 4d(d + 3)(d2 - 3d + 9)
The graph of a function hits the axis and turns around.
What is the factor is repeated an even number of times.
Subtract the following polynomials: (34 + 8x3 - 9x2) - (3x3 + 10x2 - 4x - 4)
(34 + 8x3 - 9x2) - (3x3 + 10x2 - 4x - 4) = 5x3 - 19x2 + 4x + 38
Expand the following binomial: (y + 3)3
(y + 3)3 = y3 + 9y2 + 27y + 27
Divide the following polynomials: (-y2 + 2y3 + 25) / (y - 3)
(-y2 + 2y3 + 25) / (y - 3) = 2y2 + 5y + 15 + (70/y-3)
Factor the following polynomial by grouping: 4x3 - 8x2 + 9x - 18
4x3 - 8x2 + 9x - 18 = (x - 2) (4x2 + 9)
Is (x + 2) a factor of P(x) = -2x3 + 7x2 - 3x - 9 ?
(x + 2) is NOT a factor of P(x) = -2x3 + 7x2 - 3x - 9, because the remainder after synthetic division is NOT zero.
If h(x) = x2 - 6x + 3, find h(-2)
If h(x) = x2 - 6x + 3, h(-2) = (-2)2 - 6(-2) + 3 = 19
Multiply the following polynomials: (a - 3) (2 - 5a + a2)
(a - 3) (2 - 5a + a2) = a3 - 8a2 + 17a - 6
Is (x - 4) a factor of x3 - 4x2 + 3x - 5 ?
Since x3 - 4x2 + 3x - 5 / (x - 4) does not have a remainder of 0, (x - 4) is not a factor of x3 - 4x2 + 3x - 5
Factor the polynomial completely: 5x6 -25w4 +30w2
5x6 -25w4 +30w2 factored is 5w2(w4 -5w2 + 6) =
5w2(w2 - 3)(w2 - 2)
Find the zeros of: f(x) = x4 - 10x2 + 9
The zeros of f(x) = x4 - 10x2 + 9 are x = 3, x = -3, x = 1, x = -1
Use a graphing calculator to find the zeros of f(x) = 2x2 + 5x + 3
The zeros of f(x) = 2x2 + 5x + 3 are -1 and -3/2
Whiteboard Daily Double!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Wager the amount of money you would like to gamble on this problem. Mrs. Crean will write down the question on the board. Good Luck!
The answer is ...
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 x3 - 2x2 - 5x + 6
Whiteboard Daily Double!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Mrs. Crean will write down the question on the white board - make your wagers now.
And the answer is ...
Find all the roots of P(x) = x4 - x3 + 7x2 - 9x - 18 by using the Rational Root Theorem (ie. find all the possible roots p/q)
The roots of P(x) = x4 - x3 + 7x2 - 9x - 18 are: -1, 2, 3i, and -3i