To solve the expression [tex]\((z - 5)(z + 3)\)[/tex], we can expand it using the distributive property (also known as the FOIL method, which stands for First, Outer, Inner, Last). Here's the detailed step-by-step process:
1. First: Multiply the first terms of each binomial.
[tex]\[
z \cdot z = z^2
\][/tex]
2. Outer: Multiply the outer terms of each binomial.
[tex]\[
z \cdot 3 = 3z
\][/tex]
3. Inner: Multiply the inner terms of each binomial.
[tex]\[
-5 \cdot z = -5z
\][/tex]
4. Last: Multiply the last terms of each binomial.
[tex]\[
-5 \cdot 3 = -15
\][/tex]
Now, combine all these results together:
[tex]\[
z^2 + 3z - 5z - 15
\][/tex]
Next, combine like terms:
[tex]\[
z^2 + (3z - 5z) - 15
\][/tex]
Simplify the expression in the parentheses:
[tex]\[
3z - 5z = -2z
\][/tex]
So, the expanded form of [tex]\((z - 5)(z + 3)\)[/tex] is:
[tex]\[
z^2 - 2z - 15
\][/tex]
Hence, the simplified expression is also:
[tex]\[
z^2 - 2z - 15
\][/tex]
So, the final result of expanding and simplifying [tex]\((z - 5)(z + 3)\)[/tex] is:
[tex]\[
z^2 - 2z - 15
\][/tex]
This can be concluded with:
[tex]\[
(z^2 - 2z - 15, z^2 - 2z - 15)
\][/tex]
