#### 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.8The student willinvestigateanddescribethe 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*n*degree polynomial equation has exactly^{th}*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 *n*^{th} 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.8_{1} Describe
the relationship among solutions of an equation, zeros of a function,
x-intercepts of a graph, and factors of a polynomial expression.

All.8_{2} Define
a polynomial function, given its zeros.

· AII.8_{1} Define
a polynomial function in factored form, given its zeros.

All.8_{3} Determine
a factored form of a polynomial expression
from the x-intercepts of the graph of its corresponding function.

· AII.8_{2} Determine
a factored form of a polynomial expression from the *x*-intercepts of the graph of its
corresponding function.

All.8_{4} For a function, identify zeros of multiplicity greater than 1 and describe the effect of those zeros on
the graph of the function.

· AII.8_{3} 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.8_{5}
Given a polynomial equation, determine
the number of real solutions and non-real solutions.

· AII.8_{4} 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*