Algebraic Representation
A Polynomial degree of n is a function of the form
A polynomial must have a degree greater than or equal to zero.
EXAMPLE:
a) f(x)=(3x^4)-x+6 is a polynomial of degree 4 with leading coefficient 3.
b) f(x)=(8x^-6)-5x+7 is not a polynomial because one of terms has a negative degree.
c) f(x)=6x-(3x^4) is a polynomial of degree 4 with a leading coefficient of -3.
Note that the degree of the polynomial is the greatest power that x is raised to and the leading coefficient is the coefficient of that power no matter what order the terms are in.
EXAMPLE:
a) f(x)=(3x^4)-x+6 is a polynomial of degree 4 with leading coefficient 3.
b) f(x)=(8x^-6)-5x+7 is not a polynomial because one of terms has a negative degree.
c) f(x)=6x-(3x^4) is a polynomial of degree 4 with a leading coefficient of -3.
Note that the degree of the polynomial is the greatest power that x is raised to and the leading coefficient is the coefficient of that power no matter what order the terms are in.
Finding Roots
Knowing how to find the zeroes of a polynomial function are important to graph it All of these statements mean the same thing.
To find the roots of a polynomial you must put polynomial in factored form then set each term in factored form equal to zero.The number of times a specific root is a solution to the equation f(x)=0 is its multiplicity.
- "find the zeroes of f(x)"
- "find the roots of f(x)"
- "find all the x-intercepts of the graph of f(x)"
- "find all the solutions to f(x)=0"
To find the roots of a polynomial you must put polynomial in factored form then set each term in factored form equal to zero.The number of times a specific root is a solution to the equation f(x)=0 is its multiplicity.
Graphing Polynomials
Graphs are able to show a polynomial's end behavior, x-intercepts, y-intercepts, and behavior at roots.
END BEHAVIOR:
The end behavior of any polynomial is identical to the end behavior of it's parent polynomial(even or odd).
Below are graph for some parent polynomial functions:
END BEHAVIOR:
The end behavior of any polynomial is identical to the end behavior of it's parent polynomial(even or odd).
Below are graph for some parent polynomial functions:
Y-INTERCEPT:
The y-intercept of a polynomial is where the graph crosses the y-axis. It can be easily found be evaluating the function at the point x=0.
X-INTERCEPT:
x-intercepts are another name for the roots or zeroes of a polynomial and are found by finding the roots or zeroes
BEHAVIOR AT ROOTS:
This is what happens whenever a polynomial touches the x-axis. it can either bounce off or pass through. the two cases are shown below.
The y-intercept of a polynomial is where the graph crosses the y-axis. It can be easily found be evaluating the function at the point x=0.
X-INTERCEPT:
x-intercepts are another name for the roots or zeroes of a polynomial and are found by finding the roots or zeroes
BEHAVIOR AT ROOTS:
This is what happens whenever a polynomial touches the x-axis. it can either bounce off or pass through. the two cases are shown below.