Certainly! Let's solve the given problem step-by-step:
### Step 1: Express [tex]\( b \)[/tex] in terms of [tex]\( a \)[/tex]
We are given the equation:
[tex]\[ 3a + b - 5 = 0 \][/tex]
Rearranging this to express [tex]\( b \)[/tex] in terms of [tex]\( a \)[/tex], we get:
[tex]\[ b = 5 - 3a \][/tex]
### Step 2: Substitute [tex]\( b = 5 - 3a \)[/tex] into the expression [tex]\( 27a^3 + b^3 - 125 \)[/tex]
Now, we substitute [tex]\( b \)[/tex] into the expression [tex]\( 27a^3 + b^3 - 125 \)[/tex]:
[tex]\[ 27a^3 + (5 - 3a)^3 - 125 \][/tex]
### Step 3: Expand the expression [tex]\( (5 - 3a)^3 \)[/tex]
Next, we need to simplify the expression [tex]\( (5 - 3a)^3 \)[/tex]. Using the binomial theorem:
[tex]\[ (5 - 3a)^3 = 5^3 - 3 \cdot 5^2 \cdot (3a) + 3 \cdot 5 \cdot (3a)^2 - (3a)^3 \][/tex]
[tex]\[ = 125 - 3 \cdot 25 \cdot 3a + 3 \cdot 5 \cdot 9a^2 - 27a^3 \][/tex]
[tex]\[ = 125 - 225a + 135a^2 - 27a^3 \][/tex]
Substituting this back into our original expression, we get:
[tex]\[ 27a^3 + (125 - 225a + 135a^2 - 27a^3) - 125 \][/tex]
### Step 4: Simplify the expression
Now, combining like terms, we get:
[tex]\[ 27a^3 + 125 - 225a + 135a^2 - 27a^3 - 125 \][/tex]
The [tex]\( 27a^3 \)[/tex] and [tex]\( -27a^3 \)[/tex] cancel each other out, and the [tex]\( 125 \)[/tex] and [tex]\( -125 \)[/tex] cancel each other out, leaving us with:
[tex]\[ -225a + 135a^2 \][/tex]
### Step 5: Factor the simplified expression
We can now factor the remaining expression:
[tex]\[ -225a + 135a^2 = 45a(3a - 5) \][/tex]
Thus, the factored form of the expression [tex]\( 27a^3 + b^3 - 125 \)[/tex] when [tex]\( 3a + b - 5 = 0 \)[/tex] is:
[tex]\[ 45a(3a - 5) \][/tex]
### Conclusion
The final factors of the expression [tex]\( 27a^3 + b^3 - 125 \)[/tex] when [tex]\( b = 5 - 3a \)[/tex] are:
[tex]\[ 27a^3 + (5 - 3a)^3 - 125 = 45a(3a - 5) \][/tex]
