Expressions and Operations 

Equations and Inequalities 

Functions 

Statistics 

SOL Review Materials 




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

Essential Knowledge and Skills 
 Describe the relationships among solutions of an equation, zeros of a function, xintercepts of a graph, and factors of a polynomial expression.
 Define a polynomial function, given its zeros.
 Determine a factored form of a polynomial expression from the xintercepts of the graph of its corresponding function.
 For a function, identify zeros of multiplicity greater than 1 and describe the effect of those zeros on the graph of the function.
 Given a polynomial equation, determine the number of real solutions and nonreal solutions.

Essential Understandings 
 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 xintercept for the graph of y = f(x).


Vertical Articulation 
 SOL A.2  perform operations on polynomials ‐ c) factor first and second degree binomials/trinomials (1 or 2 variables)

Instructional Materials 
 VDOE ESS Lesson: Factors, Zeros, and Solutions  Functions (PDF)  Exploring relationships among factors, zeros, and solutions (Word)
 Notes: Quadratics  Connecting the Dots (ppt)
 Activity: Making Connections (zip)

Real World Lessons 

Practice 
 VDOE ESS: Zeros and Factors Worksheet (pdf)
 Worksheet: Connections Between Graphs and Factors (doc)

SelfAssessments 
 SOL MiniQuiz: AII.8  Solutions, Zeros, and Factors

Skills 
 Mathematics Assessment Project: A17:Cubic Graph  Apprentice  In this task, you will look at the properties of a cubic equation.
 JMU Pivotal Question:
Relationships Among Zeros, Intercepts, Factors, and Solutions (doc)

Videos 

Explore
Learning 
 Polynomials and Linear Factors  Create a polynomial as a product of linear factors. Vary the values in the linear factors to see how their connection to the roots of the function.

Websites 

Common Core
Standards 

