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Birthdays 6th Period

Birthdays 6th Period. Rachel Mcclure – October 22. Warm-up. If one solution to a polynomial is –4 + 2i what is another solution? Find all possible rational zeros for: f(x) = 3x 3 – 4x 2 – 3x + 6. Solutions of Polynomials. 2.5. BIG Picture.

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Birthdays 6th Period

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  1. Birthdays 6th Period • Rachel Mcclure – October 22

  2. Warm-up • If one solution to a polynomial is –4 + 2i what is another solution? • Find all possible rational zeros for: f(x) = 3x3 – 4x2 – 3x + 6

  3. Solutions of Polynomials 2.5

  4. BIG Picture • The degree of the polynomial tells us how many complex solutions we have and the most rational zeros we can have • The Rational Zero Theorem tells us POSSIBLE rational zeros • WE need to use other methods to find the actual zeros (solutions) • See steps on next Slide

  5. Finding Zeros • 1. Use Rational Zero Theorem to find all possible rational zeros • 2. Use Trick of “1” • 3. Next use synthetic division to test each zero, don’t forget multiplicity. • 4. When down to lowest level of synthetic division (2nd degree), use factoring if possible, otherwise use quadratic formula to find imaginary solutions.

  6. Find ALL roots 2x3 + 21x2 + 7x – 30 = 0 There are many, many possible roots for this polynomial because 30 has many factors. We can use a TRICK to help us find a root.

  7. Find ALL roots 2x3 + 21x2 + 7x – 30 = 0 Trick of “1” If when you add up all of the coefficients the sum = 0 then “1” is a root. yes Is “1” a root of this polynomial?

  8. Find ALL roots 2x3 + 11x2 – 7x – 6 = 0 Does the trick of one work? Do synthetic division with one in the box to find the depressed polynomial. 2x2 + 13x + 6 Find the remaining roots. X = 1, -6, -1/2

  9. Find the solutions and/or zeros: f(x) = x5 + x3 + 2x2 – 12x + 8 Does the trick of one work? (x - 1)(x4 + x3 + 2x2 + 4x - 8) Does the trick of one work again? (x – 1)2 (x3 + 2x2 + 4x + 8) Now what? grouping (x – 1)2 (x2 + 4)(x + 2) We want zero’s. (1,0)MP2, (+ 2i,0), (-2,0) Should the imaginary ordered pair be listed there? (1,0)MP2, (-2,0)

  10. 11) Find ALL roots. x3 + 6x2 + 10x + 3 = 0 The trick of one does not work…….. Find possible roots Try one of these….. Why would we NOT try 3? (x + 3)(x2 + 3x + 1) Now what?

  11. Use Quadratic Formula to find remaining solutions Final solutions: {2, -2 + i, -2 – i} Notice x = 2 has multiplicity = 2

  12. Possible roots are ±1, ±13 The degree is 4, so there are 4 roots! Use Synthetic Division to find the roots Use Quadratic Formula to find remaining solutions Multiplicity of 2

  13. Linear Factorization Theorem • An nth – degree polynomial can be expressed as the product of a nonzero constant and n linear factors

  14. Linear Factorization • If we know the solutions, we can work backwards and find the Linear Factorization • Example:Given the solutions to a polynomial are:{-2, 3, 6, 2+3i, 2-3i} write the polynomial as a product of its factors • f(x) = (x+2)(x-3)(x-6)(x-(2+3i))(x-(2-3i)) • Also Note: If we know all the factors and multiply them together, we get the polynomial function.

  15. Homework • WS 4-4

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