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Transcript
The Fundamental Theorem of Algebra Lesson 4.5 Example 2 x 3x 5 0 Consider the solution to Note the graph No intersections with x-axis Using the solve and csolve functions Fundamental Theorem of Algebra A polynomial f(x) of degree n ≥ 1 has at least one complex zero Number of Zeros theorem Remember that complex includes reals A polynomial of degree n has at most n distinct zeros Explain how theorems apply to these graphs Constructing a Polynomial with Prescribed Zeros Given polynomial f(x) Degree = 4 Leading coefficient 2 Zeros -3, 5, i, -i Determine factored form f ( x) 2 x 3 x 5 x i x i Determine expanded form Conjugate Zeroes Theorem Given a polynomial with real coefficients P( x) an x an 1 x n n 1 ... a1 x a0 If a + bi is a zero, then a – bi will also be a zero Finding Imaginary Zeros f ( x) x x 2 x x 1 4 Given Determine all zeros Use calculator to factor Try cFactor command Use calculator to graph Use cSolve or cZeros 3 2 Application Complex numbers show up in study of electrical circuits Impedance, Z Voltage, V Current, I Can be represented by complex numbers Find missing value V Z I V 30 60i I 8 6i Z 10 5i V 10 8i Assignment Lesson 4.5 Page 307 Exercises 1 – 41 EOO 43 – 47 odd