1. Linear Factorizations Sec. 2.6b

2. First, remind me of the definition of a linear factorization… f (x) = a(x – z )(x – z )…(x – z ) An equation in the following form: 12n

3. Now, Our Practice Problems: Find all zeros of the given function, and write the function in its linear factorization. Check the graph for possible real zeros… Possibly, x = –2, x = 1, and x = 4 –21–3–55–68 –210–1010–8 1–55 40 Check and factor, using synthetic division:

4. Now, Our Practice Problems: Find all zeros of the given function, and write the function in its linear factorization. 11–55 4 1–41 1 1 0

5. Now, Our Practice Problems: Find all zeros of the given function, and write the function in its linear factorization. 41–41 404 1010

6. Now, Our Practice Problems: Find all zeros of the given function, and write the function in its linear factorization. Complete Linear Factorization:

7. Now, Our Practice Problems: The complex number z = 1 – 2i is a zero of the given function. Find the remaining zeros of the function, and write it in its linear factorization. 1 – 2i40171465 4 – 8i–12 – 16i–27 – 26i–65 45 – 16i–13 – 26i04 – 8i

8. Now, Our Practice Problems: The complex number z = 1 – 2i is a zero of the given function. Find the remaining zeros of the function, and write it in its linear factorization. 1 + 2i44 – 8i5 – 16i–13 – 26i 4 + 8i8 + 16i13 + 26i 41308 Use the quadratic formula to find the last two zeros… 1 + 2i must also be a zero!!!

9. Now, Our Practice Problems: The complex number z = 1 – 2i is a zero of the given function. Find the remaining zeros of the function, and write it in its linear factorization. Now we can write the linear factorization…

10. Now, Our Practice Problems: The complex number z = 1 – 2i is a zero of the given function. Find the remaining zeros of the function, and write it in its linear factorization.