1 Using the Fundamental Theorem of Algebra.  Talk about #56 & #58 from homework!!!  56 = has -1 as an answer twice  58 = when you go to solve x 2 +

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Presentation transcript:

1 Using the Fundamental Theorem of Algebra

 Talk about #56 & #58 from homework!!!  56 = has -1 as an answer twice  58 = when you go to solve x you end up with 2 imaginary solutions!  Directions only asked for REAL Solutions 2

Remember??? Complex numbers : a real number and an imaginary number a + bi Complex Conjugate Theorem: If a + bi is a zero/solution of the function, then a – bi is also a zero.

The Fundamental Theorem of Algebra  Recall:  the degree of the polynomial = the number of roots/solutions the polynomial has  Every polynomial of degree “n” with complex coefficients has “n” roots in the complex numbers.  This does NOT say n distinct roots  So, there may be repeated roots 4

Find all the zeros of the polynomial. (real and complex) 5 List all possible rational zeros. Narrow down list using A graphing calculator. -4, 3

6 Use synthetic division to test these zeros and factor the polynomial. The quadratic will not factor, so use the quadratic formula to solve it.

Assignment  p.369 # all 7

8 Using the Fundamental Theorem of Algebra

 Going backwards!  Last week --- given the function and ask to find zero’s  Now --- given zero’s and asked to find the function 9

Examples  Write a polynomial function of least degree that has real coefficients and a leading coefficient of 1 with the following zeros. 1. 3, -2, , i, -i 3. 1 – 2i, 3, -3 10

, -2, 5 3 as a zero is (x - 3) as a factor -2 as a zero is (x + 2) as a factor 5 as a zero os (x – 5) as a factor FOIL the first two binomials Distribute each term. Combine like terms

, i, -i 4 as a zero is (x - 4) as a factor i as a zero is (x - i) as a factor -i as a zero os (x + i) as a factor FOIL the last two binomials (multiply the complex parts first!) FOIL Combine like terms & replace i 2 with -1 Rearrange terms

– 2i, 3, -3 All imaginary factors come in conjugate pairs, so 1 + 2i is also a zero. Distribute the negative through each complex # Distribute the first two polynomials Combine like terms & replace i 2 with -1 Now distribute and FOIL

 Zero’s: 3, 6i, 2-3i  Factors = (x - 3) (x + 6i)(x - 6i) (x –(2-3i)) (x – (2+3i)) 14

Assignment  p evens 15