To simplify the rational expression \( \frac{t^2 + 4t}{32t + 8} \), we can start by factoring both the numerator and the denominator. Here’s how we can proceed:
Step 1: Factor the numerator
The numerator is \( t^2 + 4t \). We can factor out a common term, which is \( t \):
\[ t^2 + 4t = t(t + 4) \]
Step 2: Factor the denominator
The denominator is \( 32t + 8 \). We notice that both terms share a common factor of 8, so we can factor that out:
\[ 32t + 8 = 8(4t + 1) \]
Step 3: Rewrite the expression
Now that we have factored both the numerator and the denominator, we can rewrite the expression as:
\[ \frac{t(t + 4)}{8(4t + 1)} \]
Step 4: Simplifying the expression
Since there are no common factors in the numerator and the denominator that can be canceled, this is our simplified form:
\[ \frac{t(t + 4)}{8(4t + 1)} \]
Step 5: Restrictions on the variable \( t \)
When simplifying rational expressions, it’s important to consider any restrictions that might affect the variable. In this case, we must ensure that the denominator does not equal zero since division by zero is undefined.
Setting the denominator equal to zero for the constraint:
\[ 32t + 8 = 0 \]
Simplifying this equation gives:
\[ 32t = -8 \]
\[ t = -\frac{1}{4} \]
So, the restriction on the variable is that \( t \) cannot be equal to \( -\frac{1}{4} \). In conclusion, the expression is simplified to:
\[ \frac{t(t + 4)}{8(4t + 1)} \], with the restriction that \( t \neq -\frac{1}{4} \).