Sure, let's find the complete factored form of the polynomial [tex]\( 3ab^4 + 30a^3 \)[/tex].
### Step-by-Step Solution:
1. Identify Common Factors:
First, look for common factors in each term of the polynomial. Here, both terms [tex]\( 3ab^4 \)[/tex] and [tex]\( 30a^3 \)[/tex] share a common factor of [tex]\( 3a \)[/tex].
2. Factor Out the Common Factor:
We can factor [tex]\( 3a \)[/tex] out of each term:
[tex]\[
3ab^4 + 30a^3 = 3a(b^4 + 10a^2)
\][/tex]
3. Examine the Remaining Polynomial:
Now, we need to check if the polynomial inside the parenthesis, [tex]\( b^4 + 10a^2 \)[/tex], can be further factored. However, in this particular case, [tex]\( b^4 + 10a^2 \)[/tex] cannot be factored further over the real numbers.
4. Write the Final Factored Form:
Thus, the polynomial [tex]\( 3ab^4 + 30a^3 \)[/tex] factored completely is:
[tex]\[
3a(10a^2 + b^4)
\][/tex]
So, the complete factored form of the given polynomial is:
[tex]\[
3a(10a^2 + b^4)
\][/tex]
