Download Pre-Calc Section 3.5

Pre-Calculus Section 3.5
Objective: SWBAT use the Fundamental Theorem of Algebra to solve equations with
real and complex solutions.
Homework: Page 298 #13, 19, 21 and 31 – 57 odd
Daily Warm – Up
Substitute the function f ( x)  x3  2 x 2  3x  1 into the difference quotient
1. Fundamental Theorem of Algebra
a. A polynomial function has at least one zero in the set of complex numbers.
b. An nth -degree polynomial function has exactly n zeros in the set of complex
numbers, counting multiple zeros.
c. If a polynomial has only real coefficients, then any nonreal complex zeros
appear in conjugate pairs.
2. Given P( x)  x3  3x 2  x  3 find:
a. All the zeros of P.
b. The complete factorization of P.
3. Zeros and their multiplicity.
Every polynomial of degree n  1has exactly n zeros, provided that a zero of
multiplicity k is counted k times.
4. Find the complete factorization and all five zeros of the polynomial
P( x)  3x5  24 x3  48x
5. Find a polynomial P ( x ) of degree 4, with zeros i, i, 2, and  2 and with P(3)  25 .
6. Find a polynomial Q ( x) of degree 4, with zeros -2 and 0, where -2 is a zero of
multiplicity 3.
7. Find the zeros of P( x)  3x 4  2 x3  x 2  12 x  4
8. Complex zeros occur in conjugate pairs.
Of a polynomial P has real coefficients, and if the complex number z is a zero of
P, then its complex conjugate z is also a zero of P.
9. Find a polynomial P(x) of degree 3 that has integer coefficients and zeros
1
2 and (3  i ) .
10. Without factoring, determine how many positive real zeros, negative real zeros, and
imaginary zeros the following polynomial could have:
P( x)  x 4  6 x3  12 x 2  14 x  24