College Algebra, 2013a

3.4 Complex Zeros and the Fundamental Theorem of Algebra 295
3.4.1 Exercises
In Exercises 1 - 10, use the given complex numbers z and w to find and simplify the following.
Write your answers in the form a + bi.
ˆ z + w ˆ zw ˆ z 2
ˆ 1
z
ˆ
z
w
ˆ w
z
ˆ z ˆ zz ˆ (z) 2
1. z =2+3i, w =4i 2. z =1+i, w = −i
3. z = i, w = −1+2i 4. z =4i, w =2− 2i
5. z =3− 5i, w =2+7i 6. z = −5+i, w =4+2i
7. z = √ 2 − i √ 2, w = √ 2+i √ 2 8. z =1− i √ 3, w = −1 − i √ 3
9. z = 1 √ √ √ √ √ √
3
3
2 2
2 2
2 + 2 i, w = −1 2 + 2 i 10. z = − 2 + 2 i, w = − 2 − 2 i
In Exercises 11 - 18, simplify the quantity.
11. √ −49 12. √ −9 13. √ −25 √ −4 14. √ (−25)(−4)
15. √ −9 √ −16 16. √ (−9)(−16) 17. √ −(−9) 18. − √ (−9)
We know that i 2 = −1 which means i 3 = i 2 · i =(−1) · i = −i and i 4 = i 2 · i 2 =(−1)(−1) = 1. In
Exercises 19 - 26, use this information to simplify the given power of i.
19. i 5 20. i 6 21. i 7 22. i 8
23. i 15 24. i 26 25. i 117 26. i 304
In Exercises 27 - 48, find all of the zeros of the polynomial then completely factor it over the real
numbers and completely factor it over the complex numbers.
27. f(x) =x 2 − 4x + 13 28. f(x) =x 2 − 2x +5
29. f(x) =3x 2 +2x + 10 30. f(x) =x 3 − 2x 2 +9x − 18
31. f(x) =x 3 +6x 2 +6x + 5 32. f(x) =3x 3 − 13x 2 +43x − 13