1. 7.4 Remainder and Factor Theorems 7.5 Roots and Zeros
Algebra II w/ trig

2. 2 Methods for Polynomial Division can be used to find a quotient and remainder:
Long division: will work for divisors of any degree
Synthetic Division: is quicker, but only will work for divisors of the form x+k
Long Division: synthetic division:

3. Or you can use Synthetic Substitution:
If f(x) = -16t2 + 74t + 5
Find f(3):

4. I. REMAINDER THEOREM: If a polynomial f(x) is divided by (x-c), the remainder is f(c).
A. Using synthetic substitution(use when degree is greater than 2) to find f(-3) : if

5. II. FACTOR THEOREM: A polynomial f(x) has a factor (x-k) if and only if f(k)=0, so if the remainder is zero.
A. Show that (x+5) is a factor of Then find the remaining factor(s) of the polynomial.

6. B. Given a polynomial and one of its factors, find the remaining factors of the polynomials. 1.

7. 2.

8. 3.

9. 4.

10. 7.5 Roots and Zeros
FUNDAMENTAL THEOREM OF ALGEBRA: If f(x) is a polynomial with positive degree, then f(x) has at least one root.
In general:
Degree = # of solutions, roots, zeros (but sometimes the same solution can happen more than one (double root - (x+2)2 ; x = -2)
Imaginary solutions always occur in pairs: If (a+bi) is a solution, then automatically we have (a – bi) is a solution as well.

11. I. Given a function and one of its zeros, find the remaining zeros of the functions.

12. B.

13. C.

14. D.

15. E.

16. II. Write a polynomial equation with the given roots.
A. 6, 2i B. 1, 1+i

17. C. -2, 2+3i D.