Factor completely: $3w^4 + 2w^3 + 2w^2 + 3w^5$
Understand the Problem
The question asks to factor the polynomial completely. The method is to rearrange it in descending order and then factor the polynomial $3w^5 + 3w^4 + 2w^3 + 2w^2$.
Answer
$w^2(w+1)(3w^2+2)$
Answer for screen readers
$w^2(w+1)(3w^2+2)$
Steps to Solve
- Rearrange in descending order
Rearrange the polynomial in descending order of powers of $w$:
$3w^5 + 3w^4 + 2w^3 + 2w^2$
- Factor out the greatest common factor
The greatest common factor (GCF) of the terms is $w^2$. Factor $w^2$ out of the polynomial.
$w^2(3w^3 + 3w^2 + 2w + 2)$
- Factor by grouping
Group the terms inside the parenthesis in pairs:
$w^2[(3w^3 + 3w^2) + (2w + 2)]$
- Factor out the GCF from each group
Factor out $3w^2$ from the first group and $2$ from the second group:
$w^2[3w^2(w + 1) + 2(w + 1)]$
- Factor out the common binomial
Notice that $(w + 1)$ is a common factor. Factor it out:
$w^2(w + 1)(3w^2 + 2)$
The polynomial is now completely factored.
$w^2(w+1)(3w^2+2)$
More Information
The factored form of the polynomial $3w^4 + 2w^3 + 2w^2 + 3w^5$ is $w^2(w+1)(3w^2+2)$.
Tips
A common mistake is not factoring out the greatest common factor in the beginning. Also, mistakes can be made during the grouping and factoring steps if not done carefully. Another common mistake is not rearranging the polynomial in descending order, which can make the grouping step more difficult to recognize.
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