Download Precalculus – Section 4

Precalculus – Section 2.5
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 – 3i)(x + 3i)(x + 5)
Example:
Use the quadratic formula to find the zeros for: f(x) = 2x2 + 5x + 6
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
Practice:
Write a polynomial function in standard form with real coefficients whose zeros are -1 – 2i and -1 + 2i.
Practice:
Write a polynomial function in standard form with real coefficients whose zeros are -1, 2 and 1 – i.
Practice:
Write a polynomial function in standard form with real coefficients whose zeros and multiplicities are
1 (multiplicity 2); –2(multiplicity 3)