Math - 2017-18

AII.8 and *AII.8 - Solutions, Relationships, Multiplicity

AII.8    The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.

AII.8  The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.


Bloom's Level:  Analyze

Adopted: 2009

BIG IDEAS

  • I can determine the dimensions of containers to maximize volume for lowest cost, increase profits and minimize expenses for advertising and manufacturing, and find the optimal height for a basketball throw.
  • I will be able to understand the relationships between zeroes of functions, solutions of equations, and x-intercepts of graphs which will simplify factoring polynomial expressions and make solving polynomial equations easier.


UNDERSTANDING THE STANDARD

  • The Fundamental Theorem of Algebra states that, including complex and repeated solutions, an nth degree polynomial equation has exactly n roots (solutions).
  • The following statements are equivalent:
    • k is a zero of the polynomial function f;
    • (x – k) is a factor of  f(x);
    • k is a solution of the polynomial equation f(x) = 0; and
    • k is an x-intercept for the graph of  y = f(x).

·  The Fundamental Theorem of Algebra states that, including complex and repeated solutions, an nth degree polynomial equation has exactly n roots (solutions).

·  Solutions of polynomial equations may be real, imaginary, or a combination of real and imaginary.

·  Imaginary solutions occur in conjugate pairs.

·  Given a polynomial function f(x), the following statements are equivalent for any real number k, such that f(k) = 0:

­  k is a zero of the polynomial function f(x) located at (k, 0);

­  k is a solution or root of the polynomial equation f(x) = 0;

­  the point (k, 0) is an x-intercept for the graph of polynomial
f(x) = 0; and

­  (x – k) is a factor of polynomial f(x).

·  Polynomial equations may have fewer distinct roots than the order of the polynomial.  In these situations, a root may have “multiplicity.”  For instance, the polynomial equation  has two identical factors, , and one other factor, . This polynomial equation has two distinct, real roots, one with a multiplicity of 2.


ESSENTIALS

Solutions/Relationships/ Multiplicity

All.81  Describe the relationship among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.

All.82  Define a polynomial function, given its zeros.

·  AII.81  Define a polynomial function in factored form, given its zeros.

All.83  Determine a factored form of a polynomial expression  from the x-intercepts of the graph of its corresponding function.

·  AII.82  Determine a factored form of a polynomial expression from the x-intercepts of the graph of its corresponding function.

All.84  For a function, identify zeros of multiplicity greater than 1 and describe the effect of those zeros on the graph of the function.

·  AII.83  For a function, identify zeros of multiplicity greater than 1 and describe the effect of those zeros on the graph of the function.

Fundamental Theorem of Algebra

All.85  Given a polynomial equation, determine the number of real solutions and non-real solutions.

·  AII.84  Given a polynomial equation, determine the number and type of solutions.


KEY VOCABULARY

solutions, zeros, x-intercept, factors, polynomial expression, relationship, polynomial function, corresponding function, real solutions, non-real solutions

Updated: Oct 27, 2017