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Conjugate theorems, Descartes' rule of signs, Theorem on Upper bound and lower bound

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- 1. Theorems on Polynomial Functions - Part 2 PSHS Main Campus July 17, 2012PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 1 / 11
- 2. Review of Previous Discussion 1 Remainder Theorem PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 2 / 11
- 3. Review of Previous Discussion 1 Remainder Theorem 2 Factor Theorem PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 2 / 11
- 4. Review of Previous Discussion 1 Remainder Theorem 2 Factor Theorem 3 Rational Zeros/Roots Theorem PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 2 / 11
- 5. Review of Previous Discussion 1 Remainder Theorem 2 Factor Theorem 3 Rational Zeros/Roots Theorem 4 Fundamental Theorem of AlgebraExampleGraph the function f (x) = 2x4 + 7x3 − 17x2 − 58x − 24. PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 2 / 11
- 6. Quiz 7Given the function f (x) = 3x4 + 14x3 + 16x2 + 2x − 3. 1 How many zeroes does f (x) have? 2 List all the possible rational zeroes according to RZT. 3 Express f (x) as a product of binomial factors. 4 Graph f (x), and correctly label all intercepts. PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 3 / 11
- 7. Conjugate TheoremsFind the zeros of the following functions: 1 y = x3 − 4x − 15 PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 4 / 11
- 8. Conjugate TheoremsFind the zeros of the following functions: 1 y = x3 − 4x − 15 2 y = −x3 − 6x2 + 16 PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 4 / 11
- 9. Conjugate TheoremsSquare root conjugatesSquare Root Conjugate TheoremGiven a polynomial function f (x) with integer coeﬃcients:If: √ 1 (a + b c) is a zero of f (x), 2 a, b, c ∈ R, b, c = 0 √then (a − b c) is also a zero of f (x).Example √(−2 + 2 3) is a zero of −x3 − 6x2 + 16. √(−2 − 2 3) must also be a zero. PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 5 / 11
- 10. Conjugate TheoremsComplex conjugatesComplex Conjugate TheoremGiven a polynomial function f (x) with real coeﬃcients:If: 1 (a + bi) is a zero of f (x), 2 a, b ∈ R, b = 0then (a − bi) is also a zero of f (x).Example √−3 + i 11 is a zero of x3 − 4x − 15. 2 √−3 − i 11 must also be a zero. 2 PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 6 / 11
- 11. Theorem on Upper Bound and Lower BoundTheorem on Upper Bound and Lower BoundSuppose P (x) is a polynomial function divided by (x − r): 1 If r > 0 and all values on the quotient row are non-negative (positive and zero), then r is an upper bound. 2 If r < 0 and the values on the quotient row are alternately non-negative and non-positive, then r is a lower bound.ExampleFind the zeroes of f (x) = x3 − 6x2 + 5x + 12. PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 7 / 11
- 12. Descartes’ Rule of SignsDescartes’ Rule of SignsGiven polynomial function f (x) with real coeﬃcients and non-zeroconstant term: The number of positive real zeros of f (x) is equal to the number of variations of sign in f (x) or less than that by an even integer. PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 8 / 11
- 13. Descartes’ Rule of SignsDescartes’ Rule of SignsGiven polynomial function f (x) with real coeﬃcients and non-zeroconstant term: The number of positive real zeros of f (x) is equal to the number of variations of sign in f (x) or less than that by an even integer. The number of negative real zeros of f (x) is equal to the number of variations of sign in f (−x) or less than that by an even integer. PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 8 / 11
- 14. ExamplesDescartes’ Rule of SignsExamples 1 How many positive and negative real zeroes does f (x) = 2x4 − 11x3 + 14x2 + 9x − 18 have? PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 9 / 11
- 15. ExamplesDescartes’ Rule of SignsExamples 1 How many positive and negative real zeroes does f (x) = 2x4 − 11x3 + 14x2 + 9x − 18 have? 2x4 − 11x3 + 14x2 + 9x − 18 = (x + 1)(x − 3)(x − 2)(2x − 3) PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 9 / 11
- 16. ExamplesDescartes’ Rule of SignsExamples 1 How many positive and negative real zeroes does f (x) = 2x4 − 11x3 + 14x2 + 9x − 18 have? 2x4 − 11x3 + 14x2 + 9x − 18 = (x + 1)(x − 3)(x − 2)(2x − 3) 2 How many positive and negative real zeroes does g(x) = 2x4 − 7x3 − 9x2 − 21x − 45 have? PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 9 / 11
- 17. ExamplesDescartes’ Rule of SignsExamples 1 How many positive and negative real zeroes does f (x) = 2x4 − 11x3 + 14x2 + 9x − 18 have? 2x4 − 11x3 + 14x2 + 9x − 18 = (x + 1)(x − 3)(x − 2)(2x − 3) 2 How many positive and negative real zeroes does g(x) = 2x4 − 7x3 − 9x2 − 21x − 45 have? 2x4 − 7x3 − 9x2 − 21x − 45 = (2x + 3)(x − 5)(x2 + 3). PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 9 / 11
- 18. Exercises 1 Find the zeroes of y = 12x4 − 56x3 + 79x2 − 21x − 18. PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 10 / 11
- 19. Exercises 1 Find the zeroes of y = 12x4 − 56x3 + 79x2 − 21x − 18. 2 Factor 6x5 + x4 − 9x3 + 14x2 − 24x. PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 10 / 11
- 20. Exercises 1 Find the zeroes of y = 12x4 − 56x3 + 79x2 − 21x − 18. 2 Factor 6x5 + x4 − 9x3 + 14x2 − 24x. 3 How many negative zeroes does the function f (x) = x4 − 10x3 + 35x2 − 50x + 24 have? Find all zeroes of f (x). PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 10 / 11
- 21. Exercises 1 Find the zeroes of y = 12x4 − 56x3 + 79x2 − 21x − 18. 2 Factor 6x5 + x4 − 9x3 + 14x2 − 24x. 3 How many negative zeroes does the function f (x) = x4 − 10x3 + 35x2 − 50x + 24 have? Find all zeroes of f (x). 1 1 2 4 Solve for x: 1 − − 2 − 3 = 0. x x x PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 10 / 11
- 22. Exercises 1 Find the zeroes of y = 12x4 − 56x3 + 79x2 − 21x − 18. 2 Factor 6x5 + x4 − 9x3 + 14x2 − 24x. 3 How many negative zeroes does the function f (x) = x4 − 10x3 + 35x2 − 50x + 24 have? Find all zeroes of f (x). 1 1 2 4 Solve for x: 1 − − 2 − 3 = 0. x x x 5 Find the solution set of x in the inequality 2x3 + 5x2 − 14x − 8 ≤ 0. PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 10 / 11
- 23. Exercises 1 Find the length of the edge of a cube if an increase of 3 cm in one dimension and of 6 cm in another, and a decrease of 2 cm in the third, doubles the volume. PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 11 / 11
- 24. Exercises 1 Find the length of the edge of a cube if an increase of 3 cm in one dimension and of 6 cm in another, and a decrease of 2 cm in the third, doubles the volume. 2 The dimensions of a block of metal are 3 cm, 4 cm, and 5 cm, respectively. If each dimension is increased by the same number of centimeters, the volume of the block becomes 3.5 times its original volume. Determine how many centimeters were added to each dimension. PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 11 / 11
- 25. Exercises 1 Find the length of the edge of a cube if an increase of 3 cm in one dimension and of 6 cm in another, and a decrease of 2 cm in the third, doubles the volume. 2 The dimensions of a block of metal are 3 cm, 4 cm, and 5 cm, respectively. If each dimension is increased by the same number of centimeters, the volume of the block becomes 3.5 times its original volume. Determine how many centimeters were added to each dimension. 3 How long is the edge of a wooden cube if, after a slice of 1 cm thick is cut oﬀ from one side, the volume of the remaining solid is 100 cubic cm? PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 11 / 11

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