Math - 2018-19
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.
- 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).
· 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.
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
· 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
zeros, x-intercept, factors, polynomial expression, relationship, polynomial
function, corresponding function, real solutions, non-real solutions