Questions: Factor by grouping. 3v^3 - 2v^2 - 15v + 10

Factor by grouping. 3v^3 - 2v^2 - 15v + 10
Transcript text: Polynomials and Factoring Factoring a univariate polynomial by grouping: Problem type 2 Factor by grouping. \[ 3 v^{3}-2 v^{2}-15 v+10 \]
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Solution

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Solution Steps

To factor the polynomial \(3v^3 - 2v^2 - 15v + 10\) by grouping, we can follow these steps:

  1. Group the terms in pairs: \((3v^3 - 2v^2)\) and \((-15v + 10)\).
  2. Factor out the greatest common factor (GCF) from each pair.
  3. If the resulting binomials are the same, factor them out.
Step 1: Group the Terms

Group the polynomial \(3v^3 - 2v^2 - 15v + 10\) into two pairs: \[ (3v^3 - 2v^2) + (-15v + 10) \]

Step 2: Factor Out the Greatest Common Factor (GCF) from Each Pair

Factor out the GCF from each pair: \[ v^2(3v - 2) - 5(3v - 2) \]

Step 3: Factor Out the Common Binomial

Since \((3v - 2)\) is common in both terms, factor it out: \[ (3v - 2)(v^2 - 5) \]

Final Answer

\[ \boxed{(3v - 2)(v^2 - 5)} \]

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