5.7: Fundamental Theorem of Algebra Worksheet Key 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

Fundamental Theorem of Algebra 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Theorems Fundamental Theorem of Algebra is the number of solutions in a polynomial equation with rational coefficients is equal to the degree of the polynomial when including all complex solutions and solutions of multiplicity. That includes all real, imaginary, and multiplicity roots. Conjugate Root Theorem: All complex zeros (imaginaries and radical) come in conjugate pairs. If a zero is given, use the conjugate and GO BACKWARDS.   Every IRRATIONAL or IMAGINARY number has TWO roots 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Theorems D. To write a polynomial equation: Look and see if there are any imaginary or radical solutions. If there are, include its conjugate. List all of the factors Sum and product: The complex factors are more difficult to FOIL so you can use the idea that the complex factors will always make where the zeros multiply to make c and add to make b FOIL the remaining factors and set to f(x). 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 1 Write the simplest polynomial function with the given zeros of 4, –1, –2 and highest degree is 3. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 2 Write the simplest polynomial function with the given zeros of 2, –2, and 0 and highest degree is 3. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Your Turn Write the simplest polynomial function with the given zeros of 3, 0, –2 and highest degree is 3. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 3 Determine the LINEAR FACTORS that has 3, 1/2, and 3/2 are zeros. No fractions or decimals are accepted. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Your Turn Determine the LINEAR FACTORS that has –2, 1/4, and –1/2 are zeros. No fractions or decimals are accepted. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 4 Find a second degree polynomial function with real coefficients that has √2 as a zero. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 5 Find a fourth degree polynomial function with real coefficients that has –1, –1, and 3i are zeros. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Your Turn Find a fifth degree polynomial function with the given zeros of +1, –2 and –2i as linear factors 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 6 Write the simplest function with zeros of 2 + i and √3 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 6 Write the simplest function with zeros of 2 + i and √3 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 7 Write the simplest function with zeros of 2 and 3 + i 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 8 Write the simplest function with zeros of 1 – 3i, 3, and –2 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Your Turn Write the simplest function with zeros of 1 with a multiplicity of 3 and 2 – i 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Finding Zeros Use the rational root theorem (P(x) over Q(x)) to make a list of potential answers. DIVIDE the function using Do SYNTHETIC DIVISION or LONG DIVISION using these zeros to until you got it down to a QUADRATIC equation. You can now use the quadratic formula or factor. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 9 Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero of f. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 9 Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero of f. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 9 Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero of f. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 9 Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero of f. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 9 Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero of f. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 10 Find all the zeros of f(x) = x4 – 5x3 + 4x2 + 2x – 8 given that 1 + i is a zero of f. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Example 11 Find all the zeros of f(x) = x3 – 4x2 + 21x – 34 given that 1 + 4i is a zero of f. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Your Turn Find all the zeros of f(x) = x3 – 11x2 + 41x – 51 given that 4 – i is a zero of f. 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra

5.7: Fundamental Theorem of Algebra Assignment Worksheet 11/27/2018 3:00 PM 5.7: Fundamental Theorem of Algebra