To factor the expression [tex]\( 200t + 8t^\beta - 80f^2 \)[/tex], follow these steps:
1. Identify common factors:
- Notice that each term in the expression [tex]\( 200t + 8t^\beta - 80f^2 \)[/tex] contains either a factor of 8 or can be divided by it.
2. Factor out the common factor:
- Factor out the common factor of 8 from each term in the expression:
[tex]\[
200t + 8t^\beta - 80f^2 = 8(25t + t^\beta - 10f^2)
\][/tex]
3. Group and simplify:
- Inside the parentheses, the expression simplifies to:
[tex]\[
25t + t^\beta - 10f^2
\][/tex]
4. Factor the remaining expression (if possible):
- In this case, the expression inside the parentheses does not have any further common factors or standard factorizations. Therefore, it remains as is so the final factored form of the expression is:
[tex]\[
8(25t + t^\beta - 10f^2)
\][/tex]
Hence, the fully factored form of [tex]\( 200t + 8t^\beta - 80f^2 \)[/tex] is:
[tex]\[
8(-10f^2 + 25t + t^\beta)
\][/tex]
(Note: The given choices [tex]\( -8 t(t-5)(t+5) \)[/tex], [tex]\( 8 t(t-5)^2 \)[/tex], and [tex]\( 8 t(t-5)(t+5) \)[/tex] are not relevant here, because they apply to different scenarios where the expressions are set particularly for quadratic forms in [tex]\( t \)[/tex]. In this factorization, we do not need to use [tex]\( t(t-5) \)[/tex] or [tex]\( t+5 \)[/tex]).
