Questions: Completely factor the expression by grouping, if possible. 10 y^3+45 y^2-2 y-9

Completely factor the expression by grouping, if possible. 10 y^3+45 y^2-2 y-9
Transcript text: Completely factor the expression by grouping, if possible. \[ 10 y^{3}+45 y^{2}-2 y-9 \]
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Solution

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Solution Steps

To factor the expression by grouping, we first split the expression into two groups. We then factor out the greatest common factor from each group. If the resulting binomials are the same, we can factor them out, leading to the completely factored expression.

Step 1: Group the Terms

The given expression is \(10y^3 + 45y^2 - 2y - 9\). We can group the terms as follows:

  • Group 1: \(10y^3 + 45y^2\)
  • Group 2: \(-2y - 9\)
Step 2: Factor Each Group

Factor out the greatest common factor from each group:

  • From Group 1: \(10y^3 + 45y^2 = 5y^2(2y + 9)\)
  • From Group 2: \(-2y - 9 = -1(2y + 9)\)
Step 3: Factor by Grouping

Since both groups contain the common factor \((2y + 9)\), we can factor it out: \[ 5y^2(2y + 9) - 1(2y + 9) = (2y + 9)(5y^2 - 1) \]

Final Answer

The completely factored expression is: \[ \boxed{(2y + 9)(5y^2 - 1)} \]

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