Factor completely: 35z^3 + 42z^2 + 40z + 48.
Understand the Problem
The question asks to factor the given cubic polynomial completely. The common technique for such problems is to use factoring by grouping, synthetic division, or rational root theorem to find one of the factors, then complete the factorization. We need to find the factors of 35z^3 + 42z^2 + 40z + 48.
Answer
$(5z + 6)(7z^2 + 8)$
Answer for screen readers
$(5z + 6)(7z^2 + 8)$
Steps to Solve
- Factor by Grouping
We will attempt to factor by grouping. Split the polynomial into two groups: $(35z^3 + 42z^2) + (40z + 48)$.
- Factor out the Greatest CommonFactor(GCF) from each group
From the first group $(35z^3 + 42z^2)$, the GCF is $7z^2$. Factoring this out gives $7z^2(5z + 6)$.
From the second group $(40z + 48)$, the GCF is $8$. Factoring this out gives $8(5z + 6)$.
- Rewrite using the factored terms
The expression becomes $7z^2(5z + 6) + 8(5z + 6)$.
- Factor out the Common Binomial Factor
Notice that $(5z + 6)$ is a common factor. Factoring this out results in $(5z + 6)(7z^2 + 8)$.
- Check For Further Factorization
The quadratic term $7z^2 + 8$ cannot be factored further using real numbers because it is a sum of squares ($7z^2 = (\sqrt{7}z)^2$ and $8 = (\sqrt{8})^2$), and there is no linear term.
$(5z + 6)(7z^2 + 8)$
More Information
The polynomial has been factored into a linear term $(5z + 6)$ and an irreducible quadratic term $(7z^2 + 8)$.
Tips
A common mistake is to stop after finding the first factor or to incorrectly factor out the GCF from each group. Another mistake is to attempt to factor $7z^2 + 8$ further when it cannot be factored using real numbers.
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