Factor completely. 14s^3 - 7s^2 + 12s - 6
Understand the Problem
The question is asking to factor the polynomial expression completely: 14s^3 - 7s^2 + 12s - 6.
Answer
$$(2s - 1)(7s^2 + 6)$$
Answer for screen readers
The completely factored form of the polynomial is $$(2s - 1)(7s^2 + 6)$$
Steps to Solve
- Group the Terms
First, we can group the polynomial into two parts: $$ (14s^3 - 7s^2) + (12s - 6) $$
- Factor Out the Greatest Common Factor (GCF) from Each Group
Now, we factor out the GCF from each group:
- From the first group $14s^3 - 7s^2$, the GCF is $7s^2$.
- From the second group $12s - 6$, the GCF is $6$. So we factor: $$ 7s^2(2s - 1) + 6(2s - 1) $$
- Factor Out the Common Binomial
Both groups contain the common binomial $(2s - 1)$. We can factor it out: $$ (2s - 1)(7s^2 + 6) $$
- Check for Further Factorization
We check if $7s^2 + 6$ can be factored further. Since $7s^2 + 6$ has no real roots, we conclude that it cannot be factored further.
The completely factored form of the polynomial is $$(2s - 1)(7s^2 + 6)$$
More Information
Factoring polynomials is a crucial skill in algebra, allowing us to simplify expressions and solve equations more easily. The method of grouping is especially useful for polynomials with four or more terms.
Tips
- Not grouping the polynomial correctly.
- Forgetting to factor out the GCF from both groups.
- Assuming that the resulting quadratic can always be factored when it may not have real roots.
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