5-1 and 5-2 Review Page
5-1 Polynomial Functions
Basic Concepts
Monomial
A monomial is either a real number, a variable, or a product of real numbers and variables with whole number exponents.
Degree of a Monomial
The degree of a monomial in one variable is the exponent of the variable.
Polynomial
A polynomial is a monomial or the sum of monomials.
Degree of a Polynomial
The degree of a polynomial is the greatest degree among its monomial terms.
Polynomial Function
A polynomial in the variable x defines a polynomial function of x.
Turning Point
A turning point of the graph of a function is a point where the graph changes direction from upwards to downwards or from downwards to upwards.
End Behavior
End Behavior of the graph of a function describes the directions of the Graph as you move to the left and to the right, away from the origin.
Standard Form of a Polynomial Function
The standard form of a polynomial function arranges the terms by degree in descending numerical order.
A monomial is either a real number, a variable, or a product of real numbers and variables with whole number exponents.
Degree of a Monomial
The degree of a monomial in one variable is the exponent of the variable.
Polynomial
A polynomial is a monomial or the sum of monomials.
Degree of a Polynomial
The degree of a polynomial is the greatest degree among its monomial terms.
Polynomial Function
A polynomial in the variable x defines a polynomial function of x.
Turning Point
A turning point of the graph of a function is a point where the graph changes direction from upwards to downwards or from downwards to upwards.
End Behavior
End Behavior of the graph of a function describes the directions of the Graph as you move to the left and to the right, away from the origin.
Standard Form of a Polynomial Function
The standard form of a polynomial function arranges the terms by degree in descending numerical order.
Degree and Number of a Polynomial
You can classify a polynomial by its degree or by its number of terms. Polynomials of degrees zero through five have specific names, as shown in this table:
End Behavior and Turning Points
The degree of a polynomial function affects the shape of its graph and determines the maximum number of turning points, or places where the graph changes direction It also effects the End Behavior, or the directions of the graph to the far left and to the far right. A function is increasing when the y-values increase as the x-values increase. A function is decreasing when the y-values decrease as x-values increase. You can determine the end behavior of a polynomial function of degree n from he leading term ax^n of the standard form.5
5-2 Polynomials, Linear Factors, and Zeros
Basic Concepts
Factor Theorem
The Expression (x-a) is a linear factor of a polynomial if and only if the value of a is a root and the related polynomial function.
Multiple Zero
If a linear factor is repeated int eh complete factored form of a polynomial, the zero related to that factor is a multiple zero.
Multiplicity
The multiplicity of a zero of a polynomial function is the number of times the related linear factor is repeated in the factored form of the polynomial.
Relative Maximum
A relative maximum is the value of the function at an up-to-down turning point.
Relative Minimum
A relative minimum is the value of the function at a down-to-up turning point.
The Expression (x-a) is a linear factor of a polynomial if and only if the value of a is a root and the related polynomial function.
Multiple Zero
If a linear factor is repeated int eh complete factored form of a polynomial, the zero related to that factor is a multiple zero.
Multiplicity
The multiplicity of a zero of a polynomial function is the number of times the related linear factor is repeated in the factored form of the polynomial.
Relative Maximum
A relative maximum is the value of the function at an up-to-down turning point.
Relative Minimum
A relative minimum is the value of the function at a down-to-up turning point.
Zeros of a Polynomial
Finding the Zeros of a polynomial function will help you factor the polynomial, graph the function, and solve the related polynomial equation. For examples, if the zeros of a graph are -2,2, and 3, then the equation for that graph automatically becomes (x-2)(x+2)(x-3).
Finding the Relative Minimum/Maximum on your Calculator
Step One:
Put your Equation into "y="
Step Two:
Graph the Polynomial
Step Three:
Hit "2nd --> Trace"
Step Four:
If you are trying to find the Relative Maximum, hit maximum.
If you are trying to find the Relative Minimum, hit minimum.
Step Five:
If you are trying to find the Relative Maximum, scroll the cursor to the left of the highest point and hit Enter.
If you are trying to find the Relative Minimum, scroll he cursor to the left of the lowest point and hit Enter.
Step Six:
If you are trying to find the Relative Maximum, scroll the cursor to the right of the highest point and hit Enter.
If you are trying to find the Relative Minimum, scroll the cursor tot he right of the lower point and hit Enter.
Step Seven:
Hit Enter again, and, on the bottom of the screen, the Relative Minimum/Relative Maximum should be displayed.
Put your Equation into "y="
Step Two:
Graph the Polynomial
Step Three:
Hit "2nd --> Trace"
Step Four:
If you are trying to find the Relative Maximum, hit maximum.
If you are trying to find the Relative Minimum, hit minimum.
Step Five:
If you are trying to find the Relative Maximum, scroll the cursor to the left of the highest point and hit Enter.
If you are trying to find the Relative Minimum, scroll he cursor to the left of the lowest point and hit Enter.
Step Six:
If you are trying to find the Relative Maximum, scroll the cursor to the right of the highest point and hit Enter.
If you are trying to find the Relative Minimum, scroll the cursor tot he right of the lower point and hit Enter.
Step Seven:
Hit Enter again, and, on the bottom of the screen, the Relative Minimum/Relative Maximum should be displayed.