1 / 25

Warm Up

Find Rational Zeros. Warm Up. Lesson Presentation. Lesson Quiz. Warm-Up. Factor completely. 1. x 2 – x – 12. ( x – 4)( x + 3). ANSWER. 2. 2 x 2 – 5 x – 3. ( x – 3)(2 x + 1). ANSWER. Warm-Up. Factor completely.

geordi
Download Presentation

Warm Up

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Find Rational Zeros Warm Up Lesson Presentation Lesson Quiz

  2. Warm-Up Factor completely. 1.x2–x– 12 (x– 4)(x + 3) ANSWER 2. 2x2– 5x– 3 (x – 3)(2x + 1) ANSWER

  3. Warm-Up Factor completely. 3. Use synthetic division to divide2x3– 3x2– 18x– 8 by x– 4. 2x2 + 5x + 2 ANSWER The volume of a box is modeled byf (x) = x(x– 1)(x– 2), where x is the length in meters. What is the volume when the length is 3 meters? 4. ANSWER 6m3

  4. 4 12 1 2 6 3 Possible rational zeros:+ , + , + , + , + , + 1 1 1 1 1 1 Example 1 List the possible rational zeros of f using the rational zero theorem. a. f (x) = x3 + 2x2 – 11x + 12 Factors of the constant term: + 1, + 2, + 3, + 4, + 6, + 12 Factors of the leading coefficient:+ 1 Simplified list of possible zeros:+ 1, + 2, + 3, + 4, + 6, + 12

  5. Possible rational zeros: 2 1 5 10 1 2 5 10 2 5 1 + , + , + , + , + , + , + , + , + , + , + 4 2 2 4 2 2 4 1 1 1 1 + 10 4 1 Simplified list of possible zeros:+ 1, + 2, + 5, + 10, + , + , + 2 + 5 1 5 2 4 4 Example 1 b. f (x) = 4x4 – x3 – 3x2 + 9x – 10 Factors of the constant term: + 1, + 2, + 5, + 10 Factors of the leading coefficient:+ 1, + 2, + 4

  6. Guided Practice List the possible rational zeros of fusing the rational zero theorem. 1. f (x) = x3 + 9x2 + 23x + 15 ANSWER + 1, + 3, + 5 + 15 2. f (x) =2x3 + 3x2 – 11x – 6 1 3 + + 1, + 2, + 3, + 6, + , ANSWER 2 2

  7. List the possible rational zeros. The leading coefficient is 1 and the constant term is 20. So, the possible rational zeros are: 2 10 x = + , + , + , + , + , + 5 20 4 1 1 1 1 1 1 1 Example 2 Find all real zeros off (x) = x3 – 8x2 + 11x + 20. SOLUTION STEP 1

  8. Test x =1: 1 1 – 8 11 20 1 – 7 4 1 – 7 4 24 ↑ 1 is not a zero. Test x = –1: –1 1 –8 11 20 –1 9 –20 1 – 9 20 0 ↑ –1 is a zero Example 2 STEP 2 Test these zeros using synthetic division.

  9. ANSWER The zeros of fare –1, 4, and 5. Example 2 Because –1 is a zero of f, you can write f (x) = (x + 1)(x2 – 9x + 20). STEP 3 Factor the trinomial in f (x) and use the factor theorem. f (x) = (x + 1) (x2 – 9x + 20) = (x + 1)(x – 4)(x – 5)

  10. Guided Practice Find all real zeros of the function. 3.f (x) = x3 – 4x2 – 15x + 18 –3, 1, 6 ANSWER 4. f (x) = x3 – 8x2 + 5x+ 14 –1, 2, 7 ANSWER

  11. List the possible rational zeros of f : 1 3 + , + , + , + , + , + , , + 6 12 + 1 , + 1 2 3 4 1 1 1 1 2 1 1 2 5 2 3 ,+ , +, + , + , +, + , + 3 4 12 1 6 5 5 5 5 5 10 10 Example 3 Find all real zeros off (x) =10x4 – 11x3 – 42x2 + 7x + 12. SOLUTION STEP 1

  12. Choose reasonable values from the list above to check using the graph of the function. For f, the values 3 12 3 1 x = – , x = – , x = , andx = 2 5 2 5 are reasonable based on the graph shown at the right. Example 3 STEP 2

  13. 3 – 10 –11 –42 7 12 2 9 69 –15 39 – 4 2 21 23 10 – 26 – 3 – 4 2 –1 10 – 11 – 42 7 12 2 – 5 8 17 –12 10 – 16 – 34 24 0 ↑ 1 – is a zero. 2 Example 3 STEP 3 Check the values using synthetic division until a zero is found.

  14. 1 f (x) = x + (10x3 – 16x2 – 34x + 24) 2 1 = x + (2)(5x3 – 8x2 – 17x + 12) 2 Example 3 STEP 4 Factor out a binomial using the result of the synthetic division. Write as a product of factors. Factor 2 out of the second factor. = (2x +1)(5x3 – 8x2 – 17x +12) Multiply the first factor by2.

  15. Repeat the steps above for g(x) = 5x3 – 8x2 – 17x + 12. Any zero of g will also be a zero of f. The possible rational zeros of gare: 12 1 3 4 6 2 x = + 1, + 2, + 3, + 4, + 6, + 12, + , + , + , + , + , + 5 5 5 5 5 5 3 The graph of gshows that may be a zero. Synthetic division shows that is a zero and 5 3 5 3 g (x) = x – (5x2 – 5x – 20) 5 It follows that: f (x) = (2x + 1) g (x) Example 3 STEP 5 = (5x – 3)(x2 – x – 4). = (2x + 1)(5x – 3)(x2 – x – 4)

  16. x = –(–1) +√ (–1)2 – 4(1)(–4) 2(1) x = 1 +√17 2 Example 3 Find the remaining zeros of f by solving x2 – x – 4 = 0. STEP 6 Substitute 1 for a, –1for b, and –4 for cin the quadratic formula. Simplify. ANSWER 1 3 The real zeros of f are , , 1 + √17, and1 – √17. – 2 5 2 2

  17. 3 1 3 1 6 2 2 4 ANSWER – , , Guided Practice Find all real zeros of the function. 5. f (x) = 48x3+ 4x2 – 20x + 3 6. f (x) = 2x4 + 5x3 – 18x2 – 19x + 42 ANSWER – 2, , 1+2 √2

  18. Some ice sculptures are made by filling a mold with water and then freezing it. You are making such an ice sculpture for a school dance. It is to be shaped like a pyramid with a height that is 1 foot greater than the length of each side of its square base. The volume of the ice sculpture is 4 cubic feet. What are the dimensions of the mold? Example 4 ICE SCULPTURES

  19. 1 4 = x2(x + 1) 3 Example 4 SOLUTION Write an equation for the volume of the ice sculpture. STEP 1 Write equation. 12 = x3 + x2 Multiply each side by 3 and simplify. 0 = x3 + x2 – 12 Subtract 12 from each side.

  20. List the possible rational solutions: 1 2 3 4 6 12 + , + , + , + ,+ , + 1 1 1 1 1 1 1 1 1 0 –12 1 2 2 1 2 2 –10 Example 4 STEP 2 Test possible solutions. Only positive x-values make sense. STEP3

  21. 2 1 1 0 –12 2 6 12 1 3 6 0 ↑ 2is a solution. Check for other solutions. The other two solutions, which satisfy –3 +i √15 x2 + 3x + 6 = 0, arex = and can be discarded because they are imaginary numbers. 2 Example 4 STEP 4

  22. ANSWER The only reasonable solution isx = 2. The base of themold is2feet by2feet. The height of the mold is2 + 1 = 3feet. Example 4

  23. 7. WHAT IF? In Example 4, suppose the base of the ice sculpture has sides that are 1 foot longer than the height. The volume of the ice sculpture is 6cubicfeet. What are the dimensions of the mold? ANSWER Base side: 3 ft, height: 2 ft Guided Practice

  24. ANSWER + 1, + 2, + 4 ANSWER –2, 1, 4 Lesson Quiz 1. List the possible rational zeros of f(x) = x3 + 8x2 – x + 4. Find all real zeros of the functions 2. f(x) = x3 – 3x2 – 6x + 8.

  25. ANSWER 5 –3, 2, 2 4. The volume V of a storage shed with a triangular roof can be modeled by V = x3 + x2(6 – x). If the volume of the shed is 80 cubic feet, find x. 1 2 ANSWER 4 Lesson Quiz 3. f(x) = 2x3 – 3x2 – 17x + 30.

More Related