4.6 - Fundamental Theorem of Algebra Page 300 23) 25) 27) 29) 31) 55) 57) 59) 61) 63) 65) 67) 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

Fundamental Theorem of Algebra Section 4.6 Pre–Calculus PreAP/Dual, Revised ©2015 viet.j.dang@humble.k12.tx.us 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - 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 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 1 Write the simplest polynomial function with the given zeros of 4, –1, –2 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 2 Write the simplest polynomial function with the given zeros of 2, –2, and 0. 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Your Turn Write the simplest polynomial function with the given zeros of 3, 0, –2 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - 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. 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - 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. 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 4 Find a second degree polynomial function with real coefficients that has 𝟐 as a zero. 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 5 Find a fourth degree polynomial function with real coefficients that has –1, –1, and 3i are zeros. 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Your Turn Find a fifth degree polynomial function with the given zeros of +1, –2 and –2i 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 6 Write the simplest function with zeros of 2 + i and 𝟑 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 6 Write the simplest function with zeros of 2 + i and 𝟑 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 7 Write the simplest function with zeros of 2 and 3 + i 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Example 8 Write the simplest function with zeros of 1 – 3i, 3, and –2 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Your Turn Write the simplest function with zeros of 1 with a multiplicity of –3 and 2 – i 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - 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. 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - 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. 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - 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. 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - 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. 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - 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. 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - 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. 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - 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. 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra Assignment Page 313: 7, 8, 11-14 all, 19-23 odd, 27-55 EOO 2/28/2019 4:38 PM 4.6 - Fundamental Theorem of Algebra