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Birthdays 6th Period Rachel Mcclure – October 22

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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

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Solutions of Polynomials 2.5

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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

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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 (2 nd degree), use factoring if possible, otherwise use quadratic formula to find imaginary solutions.

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Find ALL roots 2x 3 + 21x 2 + 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.

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Find ALL roots 2x 3 + 21x 2 + 7x – 30 = 0 Trick of “1” If when you add up all of the coefficients the sum = 0 then “1” is a root. Is “1” a root of this polynomial? yesyes

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Find ALL roots 2x 3 + 11x 2 – 7x – 6 = 0 Does the trick of one work? Do synthetic division with one in the box to find the depressed polynomial. 2x 2 + 13x + 6 Find the remaining roots. X = 1, -6, -1/2

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Find the solutions and/or zeros: f(x) = x 5 + x 3 + 2x 2 – 12x + 8 Does the trick of one work? (x - 1)(x 4 + x 3 + 2x 2 + 4x - 8) Now what? Does the trick of one work again? (x – 1) 2 (x 3 + 2x 2 + 4x + 8) grouping (x – 1) 2 (x 2 + 4)(x + 2) We want zero’s.(1,0)MP 2, (+ 2i,0), (-2,0) Should the imaginary ordered pair be listed there? (1,0)MP 2, (-2,0)

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11) Find ALL roots. x 3 + 6x 2 + 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)(x 2 + 3x + 1) Now what?

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

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

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

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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.

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Homework WS 4-4

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Creating Polynomials Given the Zeros.. What do we already know about polynomial functions? They are either ODD functions They are either EVEN functions.

Creating Polynomials Given the Zeros.. What do we already know about polynomial functions? They are either ODD functions They are either EVEN functions.

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