1. Chapter 7.6 Rational Zero Theorem Standard & Honors
Algebra II Mr. Gilbert
Chapter 7.6
Rational Zero Theorem
Standard & Honors
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2. Students shall be able to
Identify the possible rational zeros of a polynomial function.
Find all the rational zeros of a polynomial function.
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3. Agenda
Warm up
Home Work
Lesson
Practice
Homework
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4. Click the mouse button or press the Space Bar to display the answers.
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Click the mouse button or press the Space Bar to display the answers.
Transparency 6

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6. Homework Review
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7. Communicate Effectively
Let f(x)=
a0xn + a1xn-1 + … + an-1x + an
The Rational Zero Theorem:
If p/q is a rational number in its simpliest form and f(p/q)=0 then p is a factor of an and q is a factor of a0.
E.g. f(x)= 2x3 + 3x2 +17x+12 and f(3/2)=0 then 3 is a factor of 12 and 2 is a factor of 2.
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8. Communicate Effectively
Let f(x)=
a0xn + a1xn-1 + … + an-1x + an
Corollary (Integral Zero Theorem:
If the coefficients are integers such that a0=1 and an0, any rational zeros of the function must be factors of an.
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9. Example 1 Identify Possible Zeros (3)
Example 2 Use the Rational Zero Theorem (5)
Example 3 Find All Zeros (5)
Note: Descartes’ Rule of Signs is not on the EOC exam.
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Lesson 6 Contents

10. List all of the possible rational zeros of
If is a rational zero, then p is a factor of 4 and q is a factor of 3. The possible factors of p are 1, 2, and 4. The possible factors of q are 1 and 3.
Answer: So,
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Example 6-1a

11. List all of the possible rational zeros of
Since the coefficient of x4 is 1, the possible zeros must be a factor of the constant term –15.
Answer: So, the possible rational zeros are 1, 3, 5, and 15.
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Example 6-1b

12. List all of the possible rational zeros of each function. a.
Answer:
Answer:
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Example 6-1c

13. Let x = the height, x – 2 = the width, and x + 4 = the length.
Geometry The volume of a rectangular solid is 1120 cubic feet. The width is 2 feet less than the height and the length is 4 feet more than the height. Find the dimensions of the solid.
Let x = the height, x – 2 = the width, and x + 4 = the length.
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Example 6-2a

14. Write the equation for volume.
Formula for volume
Multiply.
Subtract 1120.
The leading coefficient is 1, so the possible integer zeros are factors of Since length can only be positive, we only need to check positive zeros.
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Example 6-2b

15. The possible factors are 1, 2, 4, 5, 8, 10, 14, 16, 20, 28, 32, 35, 40, 56, 70, 80, 112, 140, 160, 224, 280, 560, and By Descartes’ Rule of Signs, we know that there is exactly one positive real root. Make a table and test possible real zeros. p 1 2 –8
–1120 3 –5
–1125 6
–1112 10 12
112
So, the zero is 10. The other dimensions are 10 – 2 or 8 feet and or 14 feet.
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Example 6-2c

16. Check Verify that the dimensions are correct.
Answer:
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Example 6-2d

17. Geometry The volume of a rectangular solid is 100 cubic feet
Geometry The volume of a rectangular solid is 100 cubic feet. The width is 3 feet less than the height and the length is 5 feet more than the height. Find the dimensions of the solid.
Answer:
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Example 6-2e

18. The possible rational zeros are 1, 2, 3, 5, 6, 10, 15, and 30.
Find all of the zeros of
From the corollary to the Fundamental Theorem of Algebra, we know there are exactly 4 complex roots.
According to Descartes’ Rule of Signs, there are 2 or 0 positive real roots and 2 or 0 negative real roots.
The possible rational zeros are 1, 2, 3, 5, 6, 10, 15, and 30.
Make a table and test some possible rational zeros.
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Example 6-3a

19. –15
–13 3 1 2 24 –6
–17 30 11
–19 p q
Since f (2) = 0, you know that x = 2 is a zero. The depressed polynomial is
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Example 6-3b

20. There is another zero at x = 3. The depressed polynomial is
Since x = 2 is a positive real zero, and there can only be 2 or 0 positive real zeros, there must be one more positive real zero. Test the next possible rational zeros on the depressed polynomial. 5 6 1 3
–15
–13 p q
There is another zero at x = 3. The depressed polynomial is
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Example 6-3c

21. Write the depressed polynomial.
Factor
Write the depressed polynomial.
Factor. or
Zero Product Property
There are two more real roots at x = –5 and x = –1.
Answer: The zeros of this function are –5, –1, 2, and 3.
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Example 6-3d

22. Find all of the zeros of Answer: –5, –3, 1, and 3 9/21/2018
Example 6-3e

23. Homework
See Syllabus 7.6
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