To factor the polynomial expression \(12x^3 - 9x^2 - 4x + 3\), we need to find its factored form. Let's analyze this step-by-step.
Given the polynomial:
[tex]\[12x^3 - 9x^2 - 4x + 3\][/tex]
We'll break it down to recognize common patterns and use appropriate factoring techniques:
1. Look for common factors: It's helpful to see if there is a common factor for all terms, but in this case, there aren't any.
2. Group terms: One useful strategy can be grouping terms for easier factoring. Let's rewrite and group:
[tex]\[(12x^3 - 9x^2) + (-4x + 3)\][/tex]
3. Factor each group independently:
- For the first group \(12x^3 - 9x^2\), extract the common factor:
[tex]\[3x^2(4x - 3)\][/tex]
- For the second group \(-4x + 3\), there are no common factors, so it remains as is:
[tex]\[-(4x - 3)\][/tex]
4. Recognize common binomial factor: Notice that both groups share a common binomial factor \((4x - 3)\). Therefore, we can factor that out:
[tex]\[(3x^2) (4x - 3) - 1(4x - 3)\][/tex]
5. Factor out the common binomial factor:
[tex]\[(4x - 3)(3x^2 - 1)\][/tex]
Therefore, the factored form of the polynomial \(12x^3 - 9x^2 - 4x + 3\) is:
[tex]\[
(4x - 3)(3x^2 - 1)
\][/tex]
Thus, the polynomial [tex]\(12x^3 - 9x^2 - 4x + 3\)[/tex] factored is [tex]\((4x - 3)(3x^2 - 1)\)[/tex].
