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Using the Fundamental Theorem of Algebra 6.7

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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.

Page 3: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

Warm-up

Page 4: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

Go Over Quiz

Page 5: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

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

Page 6: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

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.

Page 7: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

We know:

• the degree of the equation tells you the number of solutions

• imaginary solutions come in pairs

Page 8: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

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

Page 9: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

Example 1

State the number of solutions and tell what they are.

x2 – 14x + 49 = 0

Page 10: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

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.

Page 11: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

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

Page 12: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

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.

Page 13: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

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.

Page 14: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

Closure

Page 15: Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number

Homework

Section 6.7 Page 369 – 370 #17 – 47 every 3

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