Factor completely: 33k^5 + 18k^3 + 18k^2 + 33k^4
Understand the Problem
The question asks to factor the given polynomial completely. Factoring involves rewriting the polynomial as a product of simpler expressions.
Answer
$3k^2(k+1)(11k^2+6)$
Answer for screen readers
$3k^2(k+1)(11k^2+6)$
Steps to Solve
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Rearrange the terms Rearrange the terms in descending order of the exponent of $k$: $$33k^5 + 33k^4 + 18k^3 + 18k^2$$
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Factor out the greatest common factor (GCF) The GCF of the terms is $3k^2$. Factoring this out, we get: $$3k^2(11k^3 + 11k^2 + 6k + 6)$$
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Factor by grouping Group the terms inside the parenthesis and factor by grouping: $$3k^2[(11k^3 + 11k^2) + (6k + 6)]$$ Factor out $11k^2$ from the first group and $6$ from the second group: $$3k^2[11k^2(k + 1) + 6(k + 1)]$$ Now, factor out the common factor $(k + 1)$: $$3k^2(k + 1)(11k^2 + 6)$$
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Check for further factorization Check if any of the factors can be further factored. In this case, $(k+1)$ is linear and $(11k^2 + 6)$ cannot be factored further using real numbers since it's a sum of squares (plus a constant).
$3k^2(k+1)(11k^2+6)$
More Information
The polynomial is now completely factored into irreducible factors.
Tips
- Forgetting to factor out the GCF first. This will leave a more complex expression to factor, and you might miss some factors.
- Incorrectly factoring by grouping. Make sure to factor out the correct terms from each group and that the remaining binomial factors are the same.
- Stopping before factoring completely. Always check if the resulting factors can be factored further.
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