Today in Pre-Calculus

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

2. 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: • x = k is a solution (or root) of the equation f(x) = 0 • k is a zero of the function f. • 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.

3. 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 – 3i)(x + 3i)(x + 5) f(x) = (x2 + 3ix – 3ix – 9i2)(x + 5) f(x) = (x2 + 9)(x + 5) f(x) = x3 + 5x2 + 9x + 45 Zeros: 3i, -3i, -5 x-intercepts: -5

4. Example Use the quadratic formula to find the zeros for: f(x) = 2x2 + 5x + 6 These are called complex conjugates: a-bi and a+bi

5. 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 + 2i and 4 – i So 3 – 2i and 4 + i are also zeros. f(x)= (x – 3 – 2i)(x – 3 + 2i)(x – 4 + i)(x – 4 – i) SHORTCUT: When [x – (a + bi)] and [x – (a – bi)] are factors their product always simplifies to: x2 – 2ax + (a2 + b2)

6. Complex Conjugates f(x)= (x – 3 – 2i)(x – 3 + 2i)(x – 4 + i)(x – 4 – i) SHORTCUT: x2 – 2ax + (a2 + b2) f(x)= [x2 – 2(3)x + (32 +22)][x2 – 2(4)x + (42 + (-1)2)] f(x)= (x2 – 6x + 13)(x2 – 8x + 17) x4 – 8x3 + 17x2 –6x3 + 48x2 – 102x 13x2 – 104x + 221 f(x) = x4 – 14x3 + 78x2 – 206x + 221

7. Practice Write a polynomial function in standard form with real coefficients whose zeros are -1 – 2i and -1 + 2i. f(x)= (x +1 + 2i)(x +1 – 2i) f(x)= x2 – 2(-1)x + ((-1)2 +(-2)2) f(x)= x2 + 2x + 5

8. 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)= (x2 – x – 2)(x2 – 2x + 2) x4 – 2x3 + 2x2 –x3 + 2x2 – 2x –2x2 + 4x – 4 f(x) = x4 – 3x3 + 2x2 + 2x – 4

9. 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 – 1)(x + 2)(x + 2)(x + 2) f(x)= (x2 – 2x + 1)(x2 + 4x + 4)(x + 2) x4 + 4x3 + 4x2 –2x3 – 8x2 – 8x x2 + 4x + 4 f(x) = (x4 + 2x3 –3x2 – 4x + 4)(x + 2) f(x) = x5 + 4x4 + x3 – 10x2 – 4x + 8

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