Sure, let's factor the trinomial [tex]\(5x + 16x + 12\)[/tex] step-by-step.
1. Identify the trinomial:
The trinomial we need to factor is [tex]\(5x + 16x + 12\)[/tex].
2. Rewrite the trinomial with the x-term expanded using two factors:
To factor the expression, we look for two terms that can sum up to the middle term, [tex]\(16x\)[/tex], but their product must be equal to the product of the first and last coefficient:
[tex]\[ 5x + 16x + 12 \][/tex]
3. Rewrite using factor pairs:
Expanding the middle term [tex]\(16x\)[/tex] into two terms whose coefficients need to add up to 16 and whose product should give us the product of the first term’s and the last term’s coefficients (i.e., [tex]\(5\)[/tex] times [tex]\(12\)[/tex]):
[tex]\[ 5x + 10x + 6x + 12 \][/tex]
4. Group the terms together:
Grouping the first two terms and the last two terms in the rewritten expression:
[tex]\[ (5x + 10x) + (6x + 12) \][/tex]
5. Find the greatest common factor (GCF) of each group:
- For the first group [tex]\(5x + 10x\)[/tex], the GCF is [tex]\(5x\)[/tex]:
[tex]\[ 5x(x + 2) \][/tex]
- For the second group [tex]\(6x + 12\)[/tex], the GCF is [tex]\(6\)[/tex]:
[tex]\[ 6(x + 2) \][/tex]
6. Combine the GCFs:
Now that we have factored each part, combine the GCFs to write the factored form of the trinomial:
[tex]\[ (5x + 6)(x + 2) \][/tex]
Therefore, the factored form of the trinomial [tex]\(5x + 16x + 12\)[/tex] is:
[tex]\[ (5x + 6)(x + 2) \][/tex]
