1 / 37

Warm Up

Warm Up. Solve the following equations/inequalities. Mini Quiz. Solve the following equations/inequalities. Solving Higher Degree Polynomials. Let’s Review Synthetic Substitution. To use synthetic substitution/ division:.

vilina
Télécharger la présentation

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. Warm Up • Solvethe following equations/inequalities

  2. Mini Quiz • Solvethe following equations/inequalities

  3. Solving Higher Degree Polynomials

  4. Let’s Review Synthetic Substitution

  5. To use synthetic substitution/ division: • Write the polynomial in standard form (Insert zeros for missing terms) • List coefficients • If you have x = c, divide by c • If you have (x + c) or (x – c), use the opposite of c • Drop down the first coefficient • Multiply, add down, and repeat

  6. Using Synthetic Substitution Use synthetic division to evaluate f(x) = 2x4+ -8x2+ 5x- 7 when x = 3. Ex. EQ: How do we use Long Division and Synthetic Division?

  7. Using Synthetic Substitution Polynomial in standard form 3 Coefficients x-value SOLUTION 2x4 + 0x3 + (–8x2) + 5x + (–7) Polynomial in standard form 2 0 –8 5 –7 3• Coefficients 6 18 30 105 35 10 98 2 6 The value of f(3) is the last number you write, In the bottom right-hand corner. EQ: How do we use Long Division and Synthetic Division?

  8. Ex. 1 Evaluate 4x3+ x + 7 when x = 2 2

  9. Ex. 1 Evaluate 4x3+ x + 7 when x = 2 add down multiply by the box number 2 41

  10. Synthetic Division • We will use synthetic division to divide one polynomial by another. • Ex.  ****When we divide we are finding FACTORS!!

  11. To determine if a given binomial is a factor • Divide synthetically • If your remainder is 0, then the binonial is a factor • Ex. Is x – 3 a factor for

  12. Ex. 1 (4x3 + x + 7)  (x – 2) Set = to zero & solve for x 2

  13. Ex. 1 (4x3 + x + 7)  (x – 2) add down multiply by the box number 2 41 Remainder Quotient

  14. Ex. 2 You try… (x3 + 6x2 – x – 30)  (x – 2) 30

  15. Ex. 3 (2x3 + 7x2 – 5)  (x + 3)

  16. Classwork Do Now • Worksheet • Homework pg. 87 # 7 - 12

  17. Classwork: Complete (10 min) • Divide synthetically

  18. Higher Degree Polynomial Functions

  19. An example of a polynomial function… f(x) = 6x4 + x3 – 21x2 – 15x + 36 Leading coefficient Degree (highest exponent) (in front)

  20. An example of a polynomial function… f(x) = 6x4 + x3 – 21x2 – 15x + 36 When we solve a polynomial function, we are looking for the numbers that make the equation equal to zero. “root” “zero” and “solution” all mean the same thing

  21. From Last week… f(x) = x2 + x – 6 To solve this equation… x2 + x – 6 = 0 Set equal to zero (x + 3)(x – 2) = 0 Factor Set each factor equal to zero x + 3 = 0 x – 2 = 0 x = - 3 x = 2 solve

  22. f(x) = 6x4 + x3 – 21x2 – 15x + 36 • But with larger polynomials, solving by factoring is impractical. • Later we’ll learn new methods for solving a polynomial.

  23. How many roots? How many factors? f(x) = x3 + 2x2 – 3x + 12 f(x) = x3 + 2x2 – 3x + 12 3 3 f(x) = x5 + 14x4 – 4 f(x) = x5 + 14x4 – 4 5 5 The degree tells you the number of roots or factors the polynomial will have.

  24. Repeated roots… Ex. 1 P(x) = x2 – 10x + 25 P(x) = (x - 5)(x - 5) x = 5 x = 5 5 has a multiplicity of two If a root occurs k times, it has a multiplicity of k.

  25. Repeated roots… f(x) = x2 f(x) = (x+3)(x+3)(x+3) x = 0, 0 x = -3, -3, -3 0 has a multiplicity of two -3 has a multiplicity of three

  26. Conjugate pairs -If is a root, then is also a root. Imaginary roots always come in pairs. -If a + biis a root, then a – bi is also a root. Irrational roots always come in pairs.

  27. Our goal is to be able to find all of the roots of a polynomial.

  28. Finding Roots

  29. Fundamental Theorem of Algebra -Every polynomial will have at least one linear factor

  30. Sometimes you can find all of the roots of a polynomial without doing much work.

  31. Suppose a polynomial of degree 3 has 3 – 4i and 9 as roots. Find all of the roots. 3 Ex. 2 Roots Because the degree is 3… 9 we should have 3 roots. 3 – 4i Because one root is 3 – 4i… 3 + 4i then 3 + 4i is also a root. Imaginary roots always come in pairs.

  32. Ex. 3 You try… Suppose a polynomial of degree 6 has -2 + 5i, -i, and as roots. Find all of the roots. 6 Because the degree is 6… Roots we should have 6 roots. -2 + 5i -2 – 5i -i Imaginary roots always come in pairs. i Irrational roots always come in pairs.

  33. Ex. 4 Find all of the roots. f(x) = x2 – 2x - 8 If quadratic: (x – 4)(x + 2) factor OR quadratic formula

  34. What if it is a higher degree polynomial? • If given a root or a factor, use synthetic division with the given root/factor • Then use factoring or the quadratic formula to solve the remaining quadratic

  35. Ex. 5 Given a zero, find all the zeros P(x) = x3 – 6x2 + 13x – 20;4 We should get zero for a remainder… 20 4 -8 why? 1 -2 5 Now we have a quadratic, what can we do? Factor OR quadratic formula

  36. Ex. 5 Given a zero, find all the zeros P(x) = x3 – 6x2 + 13x – 20;4 , 12i 20 4 -8 1 -2 5

  37. Warm Up • Find ALL zeros

More Related