Linear Linear FactorizatiFactorizati

onsonsSec. 2.6bSec. 2.6b

First, remind me of the definition of aFirst, remind me of the definition of alinear factorization…linear factorization…

f f ((xx) = ) = aa((xx – – zz )( )(xx – – zz )…( )…(xx – – zz ) )

An equation in the following form:An equation in the following form:

11 22 nn

Now, Our Practice Problems:Find all zeros of the given function, and write the function in itslinear factorization.

5 4 3 23 5 5 6 8f x x x x x x Check the graph for possible real zeros…

Possibly, x = –2, x = 1, and x = 4

–2 1 –3 –5 5 –6 8

–2 10 –10 10 –8

1 –5 5 –5 4 0

Check and factor, using synthetic division:

4 3 22 5 5 5 4f x x x x x x

Now, Our Practice Problems:Find all zeros of the given function, and write the function in itslinear factorization.

5 4 3 23 5 5 6 8f x x x x x x

1 1 –5 5 –5 4

1 –4 1 –4

1 –4 1 –4 0

4 3 22 5 5 5 4f x x x x x x

3 22 1 4 4f x x x x x x

Now, Our Practice Problems:Find all zeros of the given function, and write the function in itslinear factorization.

5 4 3 23 5 5 6 8f x x x x x x

4 1 –4 1 –4

4 0 4

1 0 1 0

3 22 1 4 4f x x x x x x

22 1 4 1f x x x x x

Now, Our Practice Problems:Find all zeros of the given function, and write the function in itslinear factorization.

5 4 3 23 5 5 6 8f x x x x x x

22 1 4 1f x x x x x 2 1 0x 2 1x 1x i

2 1 4f x x x x x i x i Complete Linear Factorization:

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 itslinear factorization.

4 24 17 14 65f x x x x 1 – 2i 4 0 17 14 65

4 – 8i –12 – 16i –27 – 26i –65

4 5 – 16i –13 – 26i 04 – 8i

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 itslinear factorization.

4 24 17 14 65f x x x x

1 + 2i 4 4 – 8i 5 – 16i –13 – 26i

4 + 8i 8 + 16i 13 + 26i

4 13 08

Use the quadratic formula to find the last two zeros…

1 + 2i must also be a zero!!!

24 8 13x x

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 itslinear factorization.

4 24 17 14 65f x x x x

8 64 4 4 13

2 4x

8 144

8

8 12

8

i

312i Now we can write the linear factorization…

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 itslinear factorization.

4 24 17 14 65f x x x x

4 1 2 1 2f x x i x i 3 3

1 12 2

x i x i