Factor completely: $21z^3 - 42z^2 + 25z - 50$
Understand the Problem
The question asks us to factor the given polynomial completely. This means we need to express the polynomial as a product of simpler polynomials or factors.
Answer
$(z - 2)(21z^2 + 25)$
Answer for screen readers
$(z - 2)(21z^2 + 25)$
Steps to Solve
- Group the terms
Group the first two terms and the last two terms together.
$(21z^3 - 42z^2) + (25z - 50)$
- Factor out the greatest common factor (GCF) from each group
From the first group, the GCF is $21z^2$. From the second group, the GCF is $25$.
$21z^2(z - 2) + 25(z - 2)$
- Factor out the common binomial factor
Notice that both terms now have a common factor of $(z - 2)$. Factor this out.
$(z - 2)(21z^2 + 25)$
- Check for further factorization
The term $(z - 2)$ is linear and cannot be factored further. The term $(21z^2 + 25)$ is a sum of squares, and since there is no common factor between $21$ and $25$, it cannot be factored further using real numbers.
Therefore, the complete factorization is $(z - 2)(21z^2 + 25)$.
$(z - 2)(21z^2 + 25)$
More Information
The polynomial $21z^2 + 25$ is irreducible over real numbers because it is a sum of squares and cannot be factored further using real coefficients. It does not have real roots and will always be positive for real values of $z$.
Tips
A common mistake is to incorrectly identify the greatest common factor (GCF) when factoring by grouping. Make sure to factor out the largest possible factor from each group. Another common mistake is to stop factoring prematurely. Always check if the resulting factors can be factored further.
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