1. 7.6 Rational Zero Theorem Depressed equation

2. All the possible rational Zeros To find all the possible rational zero, take all the factors of the last number divide by factors of the lead coefficient. Find all possible rational zero. Find factors of 4: {1, -1, 2, - 2, 4,- 4} Find factors of 3 : {1, -1, 3, -3}

3. All the possible rational Zeros To find all the possible rational zero, take all the factors of the last number divide by factors of the lead coefficient. Find all possible rational zero. Find factors of 4: {1, -1, 2, - 2, 4,- 4} Find factors of 3 : {1, -1, 3, -3} Possible rational zero

4. How to find all the Rational roots Given the equation: p(x) = x 3 – 5x 2 – 22x + 56 Possible roots We solve using Synthetic substitution 1 | 1 -5 -2256

5. How to find all the Rational roots Possible roots 1 | 1 -5 -2256 1 -4-26 1-4 -26 x Not zero

6. How to find all the Rational roots Possible roots 2 | 1 -5 -2256 2 -6-56 1-3 -28 0 Zero

7. How to find all the Rational roots Possible roots 2 | 1 -5 -2256 2 -6-56 1-3 -28 0 Zero “2” Use the depressed equation X 2 – 3x – 28 = 0

8. How to find all the Rational roots Possible roots 2 | 1 -5 -2256 2 -6-56 1-3 -28 0 Zero Use the depressed equation by factoring. X 2 – 3x – 28 = 0 ( x – 7)(x + 4)Zero “7, -4” and “2”

9. Find all the Zeros Possible Roots How many Positive roots are there?

10. Find all the Zeros Possible Roots How many Positive roots are there? 0 Half the work now.

11. Find all the Zeros -1 |110 33 388 - 1 -9 - 24-14 1 9 24 14 X Not – 1, reminder is not 0

12. Find all the Zeros -2 |110 33 388 - 2 -16 - 34-8 1 8 17 4 0 – 2, reminder is 0 Use the depressed equation to find the other zeros Is – 8, still a possible Zero?

13. Find all the Zeros -2 |110 33 388 - 2 -16 - 34-8 -2 |1 8 17 4 0 -2 -12 -10 1 6 5 X -2 is a reminder only once

14. Find all the Zeros -2 |110 33 388 - 2 -16 - 34-8 -4 |1 8 17 4 0 - 4 -16 - 4 1 4 1 0 -4 is a Zero !The new depressed equation x 2 + 4x + 1= 0

15. Find all the Zeros Use the Quadratic equation x 2 + 4x + 1= 0

16. Find all the Zeros Use the Quadratic equation x 2 + 4x + 1= 0 Answers

17. Homework Page 381 # 13 – 23 odd 27, 29, 31

18. Homework Page 381 # 12 – 28 odd 30