Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use...
-
Upload
beverly-blake -
Category
Documents
-
view
224 -
download
0
Transcript of Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use...
Using the Fundamental Theorem of Algebra
6.7
Learning Targets
Students should be able to…- Use fundamental theorem of algebra to
determine the number of zeros of polynomial function.
- Use technology to approximant the real zeros of polynomial function.
Warm-up
Go Over Quiz
Review
Find zeros of functions, solutions to polynomialequations
A. If possible, factorB. If not factorable, find one root by using the
Rational Root Theorem (p/q), use synthetic division and then factor the remaining quadratic.
C. Finally, set unsolved factors to zero and solve
Using the fundamental theorem to determine the number of roots.
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the setof complex numbers.
We know:
• the degree of the equation tells you the number of solutions
• imaginary solutions come in pairs
New info:
• if an imaginary number is a zero, then it’s conjugate is also a zero
– so, if 4 + 3i is a solution, then 4 – 3i is also a solution
Example 1
State the number of solutions and tell what they are.
x2 – 14x + 49 = 0
Example 2: Solve.
• x4 + 3x3 – 8x2 – 22x – 24 = 0How many solutions are there? 4
Use what you did in 6.6 to solve for them.
List the possible rational roots.
Use your calculator to pick ones to test.
Synthetically divide until you get a quadratic
Factor or use the quadratic formula.
Writing polynomial functions from zeroes
Example 1:Write a polynomial function of least degree that has real coefficients and a leading coefficient of 1 and has the roots 3, 2, –4.
1. Rewrite the zeros as factors. 2. Multiply/Expand
Example 2
Write a polynomial function of least degree that has real coefficients and a leading coefficient of 3 and has the roots 5, 2i.
Example 3
Write a polynomial function of least degree that has real coefficients and a leading coefficient of 1 and has the roots 1 and 2 + i.
Closure
Homework
Section 6.7 Page 369 – 370 #17 – 47 every 3