Questions: Use the grouping method to factor this polynomial completely. 2x^3 + 6x^2 + 5x + 15 A. (2x^2 + 3)(x + 5) B. (2x^2 + 5)(x + 5) C. (2x^2 + 3)(x + 3) D. (2x^2 + 5)(x + 3)

Transcript text: Use the grouping method to factor this polynomial completely. \[ 2 x^{3}+6 x^{2}+5 x+15 \] A. $\left(2 x^{2}+3\right)(x+5)$ B. $\left(2 x^{2}+5\right)(x+5)$ C. $\left(2 x^{2}+3\right)(x+3)$ D. $\left(2 x^{2}+5\right)(x+3)$

Solution

Solution Steps

To factor the polynomial $2x^3 + 6x^2 + 5x + 15$ using the grouping method, we first group the terms into two pairs: $(2x^3 + 6x^2)$ and $(5x + 15)$. We then factor out the greatest common factor from each pair. For the first pair, the common factor is $2x^2$, and for the second pair, it is $5$. After factoring, we should have a common binomial factor that can be factored out, resulting in the completely factored form of the polynomial.

Step 1: Group the Terms

To factor the polynomial $2x^3 + 6x^2 + 5x + 15$ using the grouping method, we first group the terms into two pairs: $(2x^3 + 6x^2)$ and $(5x + 15)$.

Step 2: Factor Out the Greatest Common Factor

For the first group $(2x^3 + 6x^2)$, the greatest common factor is $2x^2$. Factoring it out, we get: \[ 2x^2(x + 3) \]

For the second group $(5x + 15)$, the greatest common factor is $5$. Factoring it out, we get: \[ 5(x + 3) \]

Step 3: Factor by Grouping

Now, we have: \[ 2x^2(x + 3) + 5(x + 3) \]

Notice that $(x + 3)$ is a common factor. We can factor it out: \[ (x + 3)(2x^2 + 5) \]