Today in PreCalculus Notes Fundamental Theorem of Algebra

Today in Pre-Calculus • Notes: – Fundamental Theorem of Algebra – Complex Zeros • Homework • Go over quiz

Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (real and nonreal). Some of the zeros may be repeated. The following statements about a polynomial function f are equivalent if k is a complex number: 1. x = k is a solution (or root) of the equation f(x) = 0 2. k is a zero of the function f. 3. x – k is a factor of f(x) NOTE: If k is a nonreal zero, then it is NOT an x-intercept of the graph of f.

Example Write the polynomial function in standard form, and identify the zeros of the function and the x-intercepts of its graph. f(x) = (x – 3 i)(x + 5) f(x) = (x 2 + 3 ix – 9 i 2)(x + 5) f(x) = (x 2 + 9)(x + 5) f(x) = x 3 + 5 x 2 + 9 x + 45 Zeros: 3 i, -5 x-intercepts: -5

Example Use the quadratic formula to find the zeros for: f(x) = 2 x 2 + 5 x + 6 These are called complex conjugates: a-bi and a+bi

Complex Conjugates For any polynomial, if a + bi is a zero, then a – bi is also a zero. Example: Write a standard form polynomial function of degree 4 whose zeros include: 3 + 2 i and 4 – i So 3 – 2 i and 4 + i are also zeros. f(x)= (x – 3 – 2 i)(x – 3 + 2 i)(x – 4 + i)(x – 4 – i) SHORTCUT: When [x – (a + bi)] and [x – (a – bi)] are factors their product always simplifies to: x 2 – 2 ax + (a 2 + b 2)

Complex Conjugates f(x)= (x – 3 – 2 i)(x – 3 + 2 i)(x – 4 + i)(x – 4 – i) SHORTCUT: x 2 – 2 ax + (a 2 + b 2) f(x)= [x 2 – 2(3)x + (32 +22)][x 2 – 2(4)x + (42 + (-1)2)] f(x)= (x 2 – 6 x + 13)(x 2 – 8 x + 17) x 4 – 8 x 3 + 17 x 2 – 6 x 3 + 48 x 2 – 102 x 13 x 2 – 104 x + 221 f(x) = x 4 – 14 x 3 + 78 x 2 – 206 x + 221

Practice Write a polynomial function in standard form with real coefficients whose zeros are -1 – 2 i and -1 + 2 i. f(x)= (x +1 + 2 i)(x +1 – 2 i) f(x)= x 2 – 2(-1)x + ((-1)2 +(-2)2) f(x)= x 2 + 2 x + 5

Practice Write a polynomial function in standard form with real coefficients whose zeros are -1, 2 and 1 – i. f(x)= (x + 1)(x – 2)(x – 1 + i)(x – 1 – i) f(x)= (x 2 – x – 2)(x 2 – 2 x + 2) x 4 – 2 x 3 + 2 x 2 – 2 x 2 + 4 x – 4 f(x) = x 4 – 3 x 3 + 2 x 2 + 2 x – 4

Practice Write a polynomial function in standard form with real coefficients whose zeros and multiplicities are 1 (multiplicty 2); – 2(multiplicity 3) f(x)= (x – 1)(x + 2)(x + 2) f(x)= (x 2 – 2 x + 1)(x 2 + 4 x + 4)(x + 2) x 4 + 4 x 3 + 4 x 2 – 2 x 3 – 8 x 2 – 8 x x 2 + 4 x + 4 f(x) = (x 4 + 2 x 3 – 3 x 2 – 4 x + 4)(x + 2) f(x) = x 5 + 4 x 4 + x 3 – 10 x 2 – 4 x + 8

Homework • Pg. 234: 1 -11 odd, 13 -20 all