Factor completely: 21w^3 - 27w^2 + 35w - 45
Understand the Problem
The question asks to factor the given polynomial completely. We will use factoring by grouping to approach this problem.
Answer
$(7w - 9)(3w^2 + 5)$
Answer for screen readers
$(7w - 9)(3w^2 + 5)$
Steps to Solve
- Group the terms
Group the first two terms and the last two terms together.
$(21w^3 - 27w^2) + (35w - 45)$
- Factor out the greatest common factor (GCF) from each group
From the first group, the GCF is $3w^2$. From the second group, the GCF is $5$.
$3w^2(7w - 9) + 5(7w - 9)$
- Factor out the common binomial factor
Notice that both terms now have a common factor of $(7w - 9)$. Factor this out.
$(7w - 9)(3w^2 + 5)$
- Check if further factoring is possible
The term $3w^2 + 5$ cannot be factored further using real numbers, as it is a sum of squares.
$(7w - 9)(3w^2 + 5)$
More Information
The polynomial is now completely factored. Factoring by grouping is a useful technique when you have four terms and no obvious common factor for all terms.
Tips
A common mistake is not factoring out the greatest common factor (GCF) correctly in each group. Another common mistake is not recognizing the common binomial factor after factoring out the GCF from each group. Also, make sure to check if the resulting factors can be factored further.
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