Polynomial Functions
A real number, a variable, or a product of a real number and one or more variables
(What is ___________)
What is a monomial?
Relative maximum or minimum can exist when...
The graph has several turning points
Factor:
4x2-9
(2x+3)(2x-3)
Use long division to solve:
6x2+7x+2 by 2x+1
3x+2
If a+√b is a root, then a-√b is a root.
If a+bi is a root, then a-bi is a root.
(What is ___________)
What is the Conjugate Root Theorem?
Directions of a graph on far left or far right
(What is ___________)
What is end behavior?
What are the relative maximum and minimum of f(x)=x3+3x2-24x? (Use a graphing calculator)
Max: (-4,80)
Min: (2,-28)
Factor:
x2-10x+25
(x-5)2
Divide using the synthetic division method:
x3-57x+56 by x-7
x2+7x-8
A triangular array of numbers in which those at the ends of the rows are 1 and each of the others is the sum of the nearest two numbers in the row above
(What is ___________)
What is Pascal's Triangle?
Write 2x3-3x2+x5 in standard form. What is the classification of this polynomial by its degree and number of terms?
x5+2x3-3x2
Quintic Trinomial
What degree of the polynomial function that generates the data with the y-values: -2,0,6,16,30,48,70,96
2nd degree
Factor following quadratic trinomial:
6x2+11x-10
(2x+5)(3x-2)
If you divide a polynomial P(X) by (x-a), then the remainder is P(a)
(What is ___________)
What is the Remainder Theorem?
If P(x) is a polynomial of degree n ≥ 1, then P(x)= 0 has exactly n roots, including multiple roots and complex roots
(What is ___________)
What is the Fundamental Theorem of Algebra?
Consider the leading term of y=-2x2-3x+3. What is the end behavior of the graph?
down and down
The expression 'x-a' is a factor of a polynomial if and only the value of 'a' is a zero of the related polynomial function
(What is ___________)
What is Factor Theorem?
Factor by grouping:
x3+2x2-3x-6
(x+2)(x2-3)
Divide using long division:
3x4-4x3+12x2+5 by x2+1
3x2-4x+9
Remainder: 4x-4
What cubic polynomial equation has roots -4 and 2i?
P(x)= x3+4x2+4x+16
Describe the end behavior and the max number of turning points in the graph of y=x3+x+3.
down and up
max number of 2 turning points
Write a polynomial function in standard form with the given zeros:
-2,2, and 3
f(x)=x3-3x2-4x+12
Factor with the sum or difference of cubes (SOAP):
8x3+1
(2x+1)(4x2-2x+1)
Given P(x)=x5-2x3-x2+2, what is P(3)?
P(3)=182
Expand (2x-3)3 by using Pascal's Triangle and the Binomial Theorem.
8x3-36x2+54x-27