Sure, let's factor the expression [tex]\( 5u^3 + 7u^2 + 25u + 35 \)[/tex] by grouping. We'll go through the solution step-by-step.
1. Group the terms:
We'll group the expression into two pairs for ease of factoring:
[tex]\[ 5u^3 + 7u^2 \quad \text{and} \quad 25u + 35 \][/tex]
2. Factor each group separately:
- For the first group [tex]\( 5u^3 + 7u^2 \)[/tex]:
We look for common factors in [tex]\( 5u^3 \)[/tex] and [tex]\( 7u^2 \)[/tex]. The common factor here is [tex]\( u^2 \)[/tex]:
[tex]\[ 5u^3 + 7u^2 = u^2(5u + 7) \][/tex]
- For the second group [tex]\( 25u + 35 \)[/tex]:
The common factor in [tex]\( 25u \)[/tex] and [tex]\( 35 \)[/tex] is [tex]\( 5 \)[/tex]:
[tex]\[ 25u + 35 = 5(5u + 7) \][/tex]
3. Combine the factored groups:
Now we have:
[tex]\[ 5u^3 + 7u^2 + 25u + 35 = u^2(5u + 7) + 5(5u + 7) \][/tex]
4. Factor out the common binomial factor [tex]\( (5u + 7) \)[/tex]:
Notice that [tex]\( (5u + 7) \)[/tex] is a common factor in both terms [tex]\( u^2(5u + 7) \)[/tex] and [tex]\( 5(5u + 7) \)[/tex]. We can factor this out:
[tex]\[ u^2(5u + 7) + 5(5u + 7) = (5u + 7)(u^2 + 5) \][/tex]
Therefore, the factored form of the expression [tex]\( 5u^3 + 7u^2 + 25u + 35 \)[/tex] is:
[tex]\[ (5u + 7)(u^2 + 5) \][/tex]
