Polynomial Functions
A polynomial function is a type of function containing polynomials
A Polynomial Function with a degree n: a n x n + a n-1 x n-1 + ... + a 2x 2 + a 1 x + a 0
-n is a natural number
To find the degree of a polynomial:
count the changes of direction
(usually)
or
use the
A Polynomial Function with a degree n: a n x n + a n-1 x n-1 + ... + a 2x 2 + a 1 x + a 0
-n is a natural number
To find the degree of a polynomial:
count the changes of direction
(usually)
or
use the
Fundamental Theorem of Algebra
-a degree n polynomial function will have n roots
Roots: are when values of a polynomial function equal zero when they are put into the polynomial function
Imaginary Roots: are roots that can only be graphed on a complex plane
-imaginary roots always come in pairs
Roots: are when values of a polynomial function equal zero when they are put into the polynomial function
Imaginary Roots: are roots that can only be graphed on a complex plane
-imaginary roots always come in pairs
A Complex Plane: Imaginary Roots
To find the roots of a Polynomial Function Use the:
Rational Root Theorem
Rational Root Theorem: any rational roots from a polynomial are quotient of factors from the constant and the leading coefficent
Rational Roots: Factors of constant/factors of leading coefficent
Rational Roots: Factors of constant/factors of leading coefficent
Remainder and Factor Theorem
Factor Theorem: a polynomial function p(x) has a factor x-c if and only if p(c)=0
Remainder Theorem: if a polynomial p(x) is divided by x-c, then the remainder is r=p(c)
Remainder Theorem: if a polynomial p(x) is divided by x-c, then the remainder is r=p(c)
Remainder Theorem: Put the Factor into x and solve for Y
If the equation equals zero--->It is a Root!
Factor Theorem: Division
If there is no remainder with the answer, it is a Root!
There are two types of Division:
Polynomial Long Division
Synthetic Division
The only time you can use synthetic division is when the divisor is a linear binomial.
Synthetic division is easiest when the linear coefficient of a binomial is one.
Synthetic division is easiest when the linear coefficient of a binomial is one.
Constructing Polynomial Equations with Given Information
End Behavior
Refer Back to the Functions Page:
Questions:
#1:Sketch a 5 degree polynomial function
#2:Is 5 a factor of (x^2)-3x-10?
#3:Which of the following are factors of (x^3)-8(x^2)+x+42: -7, 2, -3, 5 and/or 1?
ANSWERS:
Question #1
#2:Is 5 a factor of (x^2)-3x-10?
#3:Which of the following are factors of (x^3)-8(x^2)+x+42: -7, 2, -3, 5 and/or 1?
ANSWERS:
Question #1
Answer above: answers may vary
Question #2
Question #2
Answer Above.
Question #3
Question #3
Answer above.
Synthetic Division used----if there is no remainder-->it is a factor!!
Synthetic Division used----if there is no remainder-->it is a factor!!
Common Core:
What are the Possible rational roots of x^2-2x-3? What are the rational roots? How many are there?
Answers:
Answers:
Step 1: Find the possible rational roots (factors of the constant/factors of the leading term)
Step 2: Use synthetic division to find the roots of the function (if there is no remainder---it is a root)
ANOTHER WAY BY CALCULATOR: Plug in a possible root into the function and if the answer=0 then that possible root is a root!
Step 2: Use synthetic division to find the roots of the function (if there is no remainder---it is a root)
ANOTHER WAY BY CALCULATOR: Plug in a possible root into the function and if the answer=0 then that possible root is a root!
Answer above.