Questions: Factor completely, or state that the polynomial is prime. 27 x^2+27 y^2 Select the correct choice below and fill in any answer boxes within your choice. A. 27 x^2+27 y^2= B. The polynomial is prime.

Factor completely, or state that the polynomial is prime.

27 x^2+27 y^2

Select the correct choice below and fill in any answer boxes within your choice.
A. 27 x^2+27 y^2=
B. The polynomial is prime.

Transcript text: Factor completely, or state that the polynomial is prime. \[ 27 x^{2}+27 y^{2} \] Select the correct choice below and fill in any answer boxes within your choice. A. $27 x^{2}+27 y^{2}=$ $\square$ B. The polynomial is prime.

Solution

Solution Steps

To factor the given polynomial $27x^2 + 27y^2$, we first look for a common factor in both terms. Both terms have a common factor of 27. After factoring out 27, we check if the resulting expression can be factored further. If not, we state that the polynomial is prime.

Step 1: Identify the Polynomial

We start with the polynomial $27x^2 + 27y^2$.

Step 2: Factor Out the Common Factor

Both terms in the polynomial share a common factor of 27. We can factor this out: \[ 27x^2 + 27y^2 = 27(x^2 + y^2) \]

Step 3: Analyze the Resulting Expression

The expression $x^2 + y^2$ cannot be factored further over the real numbers, as it does not have real roots. Therefore, the polynomial is not prime.