Today in Pre-Calculus

• Notes: – Fundamental Theorem of Algebra– Complex Zeros

• Homework• Go over quiz

Fundamental Theorem of AlgebraA 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) = 02. 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.

ExampleWrite 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 + 45Zeros: 3i, -3i, -5 x-intercepts: -5

ExampleUse the quadratic formula to find the zeros for:f(x) = 2x2 + 5x + 6

5 25 484

x

5 234

x

5 234ix

These are called complex conjugates:a-bi and a+bi

Complex ConjugatesFor 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 – iSo 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)

Complex Conjugatesf(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

PracticeWrite 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

PracticeWrite 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

PracticeWrite 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

Homework• Pg. 234: 1-11odd, 13-20 all