Factor completely: $18j^3+36j^2+7j+14$.
Understand the Problem
The question is asking us to fully factorize the polynomial $18j^3+36j^2+7j+14$. We can use techniques such as factoring by grouping to simplify and factorize this polynomial.
Answer
$(j+2)(18j^2+7)$
Answer for screen readers
$(j+2)(18j^2+7)$
Steps to Solve
- Group the terms
We can group the first two terms and the last two terms together: $$ (18j^3 + 36j^2) + (7j + 14) $$
- Factor out the greatest common factor (GCF) from each group
From the first group, the GCF is $18j^2$. From the second group, the GCF is $7$. Factoring these out gives us: $$ 18j^2(j+2) + 7(j+2) $$
- Factor out the common binomial factor
Notice that both terms now have a common factor of $(j+2)$. We can factor this out: $$ (j+2)(18j^2 + 7) $$
- Check if further factorization is possible
The quadratic term $18j^2 + 7$ cannot be factored further using real numbers because it is a sum of squares (plus a constant).
$(j+2)(18j^2+7)$
More Information
Factoring by grouping is a useful technique when dealing with polynomials that have four or more terms. The key is to find a grouping that allows you to factor out a common binomial factor.
Tips
A common mistake is to stop after factoring out the GCF from each group, forgetting to factor out the common binomial factor. Also, students may incorrectly identify the GCF of each group, leading to incorrect factoring. Always double-check your factored expression by expanding it to see if it matches the original polynomial.
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