Questions: Factor the following polynomial. First use a common factor with a positive coefficient, and then use a common factor with a negative coefficient. 2y^3 + 8y^4 + 6y^3

Factor the following polynomial. First use a common factor with a positive coefficient, and then use a common factor with a negative coefficient. 2y^3 + 8y^4 + 6y^3
Transcript text: Factor the following polynomial. First use a common factor with a positive coefficient, and then use a common factor with a negative coefficient. 2y^3 + 8y^4 + 6y^3
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Solution

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Solution Steps

To factor the polynomial, first identify the greatest common factor (GCF) with a positive coefficient. Then, factor out the GCF from each term of the polynomial. After that, identify a common factor with a negative coefficient and factor it out similarly.

Step 1: Identify the Polynomial

The given polynomial is

\[ 2y^3 + 8y^4 + 6y^3. \]

Step 2: Factor with a Positive Coefficient

First, we find the greatest common factor (GCF) of the polynomial with a positive coefficient. The GCF of the terms \(2y^3\), \(8y^4\), and \(6y^3\) is \(2y^3\). Factoring this out, we have:

\[ 2y^3(1 + 4y + 3) = 2y^3(8y^4 + 8y^3). \]

This simplifies to:

\[ 8y^3(y + 1). \]

Step 3: Factor with a Negative Coefficient

Next, we factor the polynomial using a common factor with a negative coefficient. By factoring out \(-8y^3\), we get:

\[ -8y^3(y + 1). \]

Final Answer

The factorizations of the polynomial are:

  1. With a positive coefficient: \(\boxed{8y^3(y + 1)}\)
  2. With a negative coefficient: \(\boxed{-8y^3(y + 1)}\)
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