To solve the problem of finding the remainders when [tex]\(P(z)\)[/tex] is divided by [tex]\( z - 1 \)[/tex] and [tex]\( z - 2 \)[/tex], given that the remainder when [tex]\(P(z)\)[/tex] is divided by [tex]\( z^2 - 3z + 2 \)[/tex] is [tex]\( 4z - 7 \)[/tex], we can proceed as follows:
### Analysis:
The polynomial [tex]\( z^2 - 3z + 2 \)[/tex] can be factored as:
[tex]\[ z^2 - 3z + 2 = (z - 1)(z - 2) \][/tex]
Given that when [tex]\(P(z)\)[/tex] is divided by [tex]\( z^2 - 3z + 2 \)[/tex], the remainder is [tex]\( 4z - 7 \)[/tex]. This means that:
[tex]\[ P(z) = (z^2 - 3z + 2)Q(z) + (4z - 7) \][/tex]
where [tex]\(Q(z)\)[/tex] is some quotient polynomial.
### Steps to Find the Remainder When Dividing by [tex]\( z - 1 \)[/tex]:
1. To find the remainder when [tex]\(P(z)\)[/tex] is divided by [tex]\( z - 1 \)[/tex], we need to evaluate the remainder [tex]\( 4z - 7 \)[/tex] at [tex]\( z = 1 \)[/tex].
2. Substitute [tex]\( z = 1 \)[/tex] into the remainder:
[tex]\[ 4(1) - 7 = 4 - 7 = -3 \][/tex]
So, the remainder when [tex]\( P(z) \)[/tex] is divided by [tex]\( z - 1 \)[/tex] is:
[tex]\[ \boxed{-3} \][/tex]
### Steps to Find the Remainder When Dividing by [tex]\( z - 2 \)[/tex]:
1. To find the remainder when [tex]\(P(z)\)[/tex] is divided by [tex]\( z - 2 \)[/tex], we need to evaluate the remainder [tex]\( 4z - 7 \)[/tex] at [tex]\( z = 2 \)[/tex].
2. Substitute [tex]\( z = 2 \)[/tex] into the remainder:
[tex]\[ 4(2) - 7 = 8 - 7 = 1 \][/tex]
So, the remainder when [tex]\( P(z) \)[/tex] is divided by [tex]\( z - 2 \)[/tex] is:
[tex]\[ \boxed{1} \][/tex]
In summary:
- The remainder when [tex]\(P(z)\)[/tex] is divided by [tex]\(z - 1\)[/tex] is [tex]\(\boxed{-3}\)[/tex].
- The remainder when [tex]\(P(z)\)[/tex] is divided by [tex]\(z - 2\)[/tex] is [tex]\(\boxed{1}\)[/tex].
