1. Chapter 7.5 Roots and Zeros Standard & Honors
Algebra II Mr. Gilbert
Chapter 7.5
Roots and Zeros
Standard & Honors
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2. Students shall be able to
Determine and find the number and type of roots for a polynomial
Find the zeros of a polynomial equation.
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3. Agenda
Warm up
Home Work
Lesson
Practice
Homework
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4. Homework Review
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5. Communicate Effectively
Theorem:
A proposition that has been or is to be proved on the basis of explicit assumptions.
The Fundamental Theorem of Algebra:
Every polynomial equation with a degree greater than 0 has at least one root in the set of complex numbers.
Complex Conjugates Theorem:
For every polynomial, if there exists a complex root a+bi then a-bi also exists.
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6. Example 1 Determine Number and Type of Roots
Example 2 Find Numbers of Positive and Negative Zeros
Example 3 Use Synthetic Substitution to Find Zeros
Example 4 Use Zeros to Write a Polynomial Function
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Lesson 5 Contents

7. Solve State the number and type of roots.
Original equation
Add 10 to each side.
Answer: This equation has exactly one real root, 10.
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Example 5-1a

8. Solve State the number and type of roots.
Original equation
Factor.
Zero Product Property or
Solve each equation.
Answer: This equation has two real roots, –8 and 6.
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Example 5-1b

9. Solve State the number and type of roots.
Original equation
Factor out the GCF.
Use the Zero Product Property. or
Subtract 6 from each side.
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Example 5-1c

10. Square Root Property
Answer: This equation has one real root at 0, and two imaginary roots at
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Example 5-1d

11. Solve State the number and type of roots.
Original equation
Factor differences of squares.
Factor differences of squares. or
Zero Product Property
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Example 5-1e

12. Solve each equation.
Answer: This equation has two real roots, –2 and 2, and two imaginary roots, 2i and –2i.
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Example 5-1f

13. Solve each equation. State the number and type of roots.
Answer: This equation has exactly one root at –3.
Answer: This equation has exactly two roots, –3 and 4.
Answer: This equation has one real root at 0 and two imaginary roots at
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Example 5-1g

14. d.
Answer: This equation has two real roots, –3 and 3, and two imaginary roots, 3i and –3i.
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Example 5-1h

15. State the possible number of positive real zeros, negative real zeros, and imaginary zeros of
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).
yes
– to +
yes
+ to – no
– to – no
– to –
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Example 5-2a

16. Since there are two sign changes, there are 2 or 0 positive real zeros
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. x 1 no
– to – no
– to –
yes
– to +
yes
+ to –
Since there are two sign changes, there are 2 or 0 negative real zeros. Make a chart of possible combinations.
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Example 5-2b

17. Answer: 2 6 4 Number of Positive Real Zeros
Number of Negative Real Zeros
Number of Imaginary Zeros
Total 2 6 4
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Example 5-2c

18. State the possible number of positive real zeros, negative real zeros, and imaginary zeros of
Answer: The function has either 2 or 0 positive real zeros, 2 or 0 negative real zeros, and 4, 2, or 0 imaginary zeros.
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Example 5-2d

19. Find all of the zeros of
Since f (x) has degree of 3, the function has three zeros. To determine the possible number and type of real zeros, examine the number of sign changes in f (x) and f (–x).
yes
yes no no no
yes
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Example 5-3a

20. The function has 2 or 0 positive real zeros and exactly 1 negative real zero. Thus, this function has either 2 positive real zeros and 1 negative real zero or 2 imaginary zeros and 1 negative real zero.
To find the zeros, list some possibilities and eliminate those that are not zeros. Use a shortened form of synthetic substitution to find f (a) for several values of a. x 1 –1 2 4 –3 –4 14
–38 –2 8
–12
Each row in the table shows the coefficients of the depressed polynomial and the remainder.
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Example 5-3b

21. Replace a with 1, b with –2, and c with 4.
From the table, we can see that one zero occurs at x = –1. Since the depressed polynomial, , is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation
Quadratic Formula
Replace a with 1, b with –2, and c with 4.
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Example 5-3c

22. Simplify.
Simplify.
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Example 5-3d

23. Answer: Thus, this function has one real zero at –1 and two imaginary zeros at and The graph of the function verifies that there is only one real zero.
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Example 5-3e

24. Find all of the zeros of
Answer:
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Example 5-3f

25. Short-Response Test Item
Write a polynomial function of least degree with integer coefficients whose zeros include 4 and 4 – i.
Read the Test Item • If 4 – i is a zero, then 4 + i is also a zero, according to the Complex Conjugate Theorem. So, x – 4, x – (4 – i), and x – (4 + i) are factors of the polynomial function.
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Example 5-4a

26. • Multiply the factors to find the polynomial function.
Solve the Test Item • Write the polynomial function as a product of its factors.
• Multiply the factors to find the polynomial function.
Write an equation.
Regroup terms.
Rewrite as the difference of two squares.
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Example 5-4b

27. Square x – 4 and replace i2 with –1.
Simplify.
Multiply using the Distributive Property.
Combine like terms.
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Example 5-4c

28. Answer: is a polynomial function of least degree with integral coefficients whose zeros are 4, 4 – i, and 4 + i.
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Example 5-4d

29. Short-Response Test Item
Write a polynomial function of least degree with integer coefficients whose zeros include 2 and 1 + i.
Answer:
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Example 5-4e

30. Homework
See Syllabus 7.5
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