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Warm Up

Warm Up. Evaluate the following polynomials. Solving Higher Degree Polynomials. Let’s Review Synthetic Division. To divide synthetically. Write the polynomial in standard form (Insert zeros for missing terms) List coefficients If you have x = c, divide by c

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Warm Up

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  1. Warm Up • Evaluate the following polynomials

  2. Solving Higher Degree Polynomials

  3. Let’s Review Synthetic Division

  4. To divide synthetically • 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

  5. Using Synthetic Substitution Use synthetic division to evaluate f(x) = 2x4+ -8x2+ 5x- 7 when x = 3. One way to evaluate polynomial functions is to usedirect substitution. Another way to evaluate a polynomialis to use synthetic substitution. EQ: How do we use Long Division and Synthetic Division?

  6. 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?

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

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

  9. Ex. 3 Evaluate 2x3+ 7x2 – 5 when x = -3)

  10. 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

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

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

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

  14. Ex. 3 (2x3 + 7x2 – 5)  (x + 3)

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

  16. Warm up • Divide synthetically

  17. Higher Degree Polynomial Functions

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

  19. 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

  20. 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

  21. 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.

  22. 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.

  23. 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.

  24. 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

  25. 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.

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

  27. Finding Roots

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

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

  30. 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.

  31. 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.

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

  33. 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

  34. 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

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

  36. Warm Up • Find ALL zeros

  37. Notes • Let P be a polynomial function in standard form with integer coefficients. If is a root of P(x), then: • p is a factor of the constant term of P • q is a factor of the leading coefficient of P The Rational Roots Theorem

  38. Discussion List all possible rational roots of P

  39. Discussion Find all roots of P:

  40. Discussion Find all roots of P:

  41. Classwork Page 104-105 #’s 1-21 EOO

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