QuestionFactor the trinomial completely. $45 s^{2}-150 s g+125 g^{2}$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $45 \mathrm{~s}^{2}-150 \mathrm{sg}+125 \mathrm{~g}^{2}=$ $\square$ (Factor completely.) B. The polynomial is prime.

Studdy Solution

STEP 1

1. We are given a trinomial of the form $ax^2 + bxy + cy^2$ .
2. The trinomial can potentially be factored into the product of two binomials.
3. We need to determine if the trinomial is factorable or if it is prime.

STEP 2

1. Identify and factor out the greatest common factor (GCF) from the trinomial.
2. Check if the resulting trinomial can be factored further into binomials.
3. Determine if the trinomial is completely factored or if it is prime.

STEP 3

Identify the greatest common factor (GCF) of the coefficients $45$ , $-150$ , and $125$ .
The GCF of these numbers is $5$ .
Factor out the GCF from the trinomial:
$45s^2 - 150sg + 125g^2 = 5(9s^2 - 30sg + 25g^2)$

STEP 4

Examine the trinomial $9s^2 - 30sg + 25g^2$ to see if it can be factored further.
Notice that this trinomial is a perfect square trinomial, which can be expressed as the square of a binomial.
$9s^2 - 30sg + 25g^2 = (3s - 5g)^2$

STEP 5

Combine the factored terms:
$5(9s^2 - 30sg + 25g^2) = 5(3s - 5g)^2$
Thus, the trinomial is completely factored as:
$\boxed{5(3s - 5g)^2}$

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