# 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.

### 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

2016 VDOE Curriculum Framework - AII.8 Understanding

·  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

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

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

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

·  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: Aug 23, 2018