Factor by grouping.

3 t Superscript 4 Baseline minus 2 t cubed minus 9 t plus 63t4−2t3−9t+6




Let's factor the expression \(3t^4 - 2t^3 - 9t + 6\) by grouping. Here's how to proceed:

### 1. Group the terms:
First, we group the terms in pairs:
\[
(3t^4 - 2t^3) - (9t - 6)
\]

### 2. Factor out the greatest common factor (GCF) from each group:
- In the first group \((3t^4 - 2t^3)\), the GCF is \(t^3\):
  \[
  t^3(3t - 2)
  \]

- In the second group \((-9t + 6)\), the GCF is \(-3\) (note the negative sign):
  \[
  -3(3t - 2)
  \]

### 3. Factor out the common binomial factor:
Now we notice that both terms have a common binomial factor \((3t - 2)\):
\[
t^3(3t - 2) - 3(3t - 2)
\]
Factor out the common binomial factor \((3t - 2)\):
\[
(3t - 2)(t^3 - 3)
\]

### Final Answer:
The expression \(3t^4 - 2t^3 - 9t + 6\) factors to:
\[
(3t - 2)(t^3 - 3)
\]

Would you like more details or have any questions?

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**Related Questions:**
1. How can you factor a cubic polynomial like \(t^3 - 3\)?
2. What would happen if the binomial factors were not the same?
3. Can you always use grouping to factor a polynomial?
4. What is the difference between factoring by grouping and other methods of factoring?
5. How can you identify the best pairs of terms for grouping?

**Tip:** When factoring by grouping, always rearrange and group terms in a way that allows you to factor out a common binomial easily.

Learn how to factor the polynomial 3t^4 - 2t^3 - 9t + 6 using the grouping method. This detailed step-by-step guide covers grouping terms, factoring out common factors, and applying the method to solve polynomial expressions.

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