Download 5.7 Notes Roots and Zeros

5.7 Notes
Roots and Zeros
Learning Goals:
I can determine the number and type of roots for a polynomial equation.
I can find the zeros of a polynomial function.
The Fundamental Theorem of Algebra
Example 1
Solve x2 + 2x – 48 = 0. State the number and type of roots.
Example 2
Solve y4 – 256 = 0. State the number and types of roots.
Example 3
Solve z2 + 12z + 20 = 0. State the number and type of roots.
Example 4
Solve a4 – 81 = 0. State the number and type of roots.
Descartes’ Rule of Signs
Find Numbers of Positive and Negative Zeros
Example 5
State the possible number of positive real zeros, negative real zeros, and imaginary zeros of
p(x) = –x6 + 4x3 – 2x2 – x – 1.
Since p(x) has degree 6, it has 6 zeros. However, some of them may be imaginary. Use Descartes’ Rule of Signs to determine the
number and type of real zeros. Count the number of changes in sign for the coefficients of p(x).
Since there are two sign changes,
there are 2 or 0 positive real zeros.
Find p(–x) and count the number of
sign changes for its coefficients.
Since there are two sign changes,
there are 2 or 0 negative real zeros.
Make a chart of possible
combinations.
Example 6
State the possible number of positive real zeros, negative real zeros, and imaginary zeros of
p(x) = x4 – x3 + x2 + x + 3.
Example 7
Find all of the zeros of f(x) = x3 – x2 + 2x + 4. First use Descartes’ Rule of Signs, and then find the zeros.
Example 8
Find all of the zeros of f(x) = x3 – 3x2 – 2x + 4. First use Descartes’ Rule of Signs, and then find the zeros.
Complex Conjugates Theorem
Example 9
Write a polynomial function with the zeros 4 and 4 – i.
Example 10
Write a polynomial function with the zeros 2 and 1 + i?