AA U1 (4.1): Characteristics of Polynomials
AAPR 3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph
of the function defined by the polynomial. (Find real and complex roots of higher degree polynomial equations using the factor
theorem, remainder theorem, rational root theorem, and fundamental theorem of algebra, incorporating complex and radical
conjugates.)
FIF 7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology
for more complicated cases.* c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and
showing end behavior.*
REFLECT
4a. Based on your results, make a generalization about the
number of times a zero occurs in the factorization of a function
and whether the graph of the function crosses or is tangent to
the x-axis at that zero.
The graph crosses the x-axis at the zero if the zero occurs
an odd number of times. The graph is tangent to the xaxis at the zero if the zero occurs an even number
of times.
YOU TRY: Sketch the graph of f (x) = (x-2)2(x +1)(x + 2)(x + 3).
The graph shows that f(x) ⟶ -∞ as x ⟶ -∞ and
f(x) ⟶ +∞ as x ⟶ +∞, and has x-intercepts -3,-2, -1,
and 2, crossing the x-axis at each negative x-intercept and
tangent to the x-axis at the positive x-intercept.
REFLECT
5a. Can you determine how many times a zero occurs in the
factorization of a polynomial function just by looking at the
graph of the function? Explain.
No; you can tell only whether a zero occurs an even or odd
number of times in the factorization.
YOU TRY: For the polynomial p(x) = 2x 3 - 7x 2 + 2x + 3,
use the Rational Zero Theorem to identify the possible rational
zeros; then, factor the polynomial completely, and sketch the
function’s graph.
Possible zeros: •+1, •+3, •+ 1/2, •+3/2;
p(x) = (x - 1)(x - 3)(2x + 1); the graph shows that
f(x) → -∞ as x → -∞ and f(x) → +∞ as x → +∞, and has
x-intercepts 1, 3, and -1/2, crossing the x-axis at each xintercept.
REFLECT
7a. How did you determine where the graph crosses the x-axis
and where it is tangent to the x-axis?
The graph crosses at any zero that occurs an odd number of
times in the factorization (i.e., at x = -3) and is tangent at
any zero that occurs an even number of times (i.e., at x =
2).
7b. How did factoring the polynomial help you graph the
function?
By setting each factor equal to zero and solving, you can
identify all the zeros of the function, which are the xintercepts of the function’s graph.
7c. How did using the Rational Zero Theorem to find one zero
help you find the other zeros?
Using synthetic substitution to find one zero gives the
coefficients of the quadratic factor, which can easily be
factored into two linear factors.