Section 4 – Honors Topic 1
Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that a polynomial
function of degree ! has exactly ! zeros, including repeated
zeros in the complex plane.
The zeros of a polynomial are also called ___________ of the
equation. Real zeros of a polynomial are points where the
graph _________________ the "-axis and # equals _________.
Real zeros are also considered ______________ zeros since
every real number is a complex number with an imaginary
part equal to _____________.
Complex zeros that contain an __________________ part not
equal to zero are not visible on the graph.
Descartes’ rule of signs can be used to find the number of
positive and negative _________ zeros of a polynomial
function.
The number of _________________ real zeros of a polynomial
function $(") is the same or less than by even numbers as the
number of changes in the sign of the coefficients.
The number of negative real zeros in a polynomial function
$(") is the same as or less than by even numbers as the
number of changes in the sign of the coefficients of the terms
of ____________.
The Rational Roots Theorem states if ((") is a polynomial with
)
integer coefficients and if is a zero of (("), then + is a factor
*
of the ________________ term of ((") and , is a factor of the
leading coefficient of ( " .
Complete the following table.
-.// − 1.2 + .4 − 1 = 6
Degree
Number of complex
zeros
Number of possible
positive real zeros
Number of possible
negative real zeros
Possible real zeros
Let’s Practice!
1.
Consider the following polynomial function.
! " = " $ − 2" ' − 5" + 6 = 0
a.
Determine the number of complex roots.
b.
Determine the number of possible positive real roots.
c.
Determine the number of negative real roots.
d.
Determine the possible rational roots.
e.
Determine the roots of the function !(").
2. Consider the graph below of the function
. " = " / + " $ − " ' + " − 2.
a. Identify the real zeros of the function.
b. Find the remaining zeros of the function.
Try It!
3. Find the zeros of the function ! " = 5" $ − " ' − 5" + 1.