Let's simplify the expression [tex]\( 4z + 5z^2 - z + z^2 \)[/tex] by combining like terms. We'll follow these steps:
1. Identify like terms: Like terms are terms that have the same variable raised to the same power.
- In this expression, the terms [tex]\( 4z \)[/tex] and [tex]\( -z \)[/tex] are like terms because they both contain [tex]\( z \)[/tex] to the first power.
- The terms [tex]\( 5z^2 \)[/tex] and [tex]\( z^2 \)[/tex] are also like terms because they both contain [tex]\( z \)[/tex] to the second power.
2. Combine like terms:
- Combine [tex]\( 4z \)[/tex] and [tex]\( -z \)[/tex]:
[tex]\( 4z - z = 3z \)[/tex]
- Combine [tex]\( 5z^2 \)[/tex] and [tex]\( z^2 \)[/tex]:
[tex]\( 5z^2 + z^2 = 6z^2 \)[/tex]
3. Rewrite the expression: After combining like terms, we get:
[tex]\[
3z + 6z^2
\][/tex]
4. Factor the expression if possible: To further simplify, we can factor the expression. Both terms [tex]\( 3z \)[/tex] and [tex]\( 6z^2 \)[/tex] share a common factor of [tex]\( 3z \)[/tex]:
[tex]\[
3z + 6z^2 = 3z (1 + 2z)
\][/tex]
So, the final simplified expression is:
[tex]\[
3z (2z + 1)
\][/tex]
The number that goes in the green box is: 3.
