Questions: Factor the trinomial completely. 25x^2(y-3)^2+35x(y-3)^2+49(y-3)^2 Select the correct choice below and fill in any answer boxes within your choice. A. 25x^2(y-3)^2+35x(y-3)^2+49(y-3)^2= B. The polynomial is prime.

Factor the trinomial completely.
25x^2(y-3)^2+35x(y-3)^2+49(y-3)^2

Select the correct choice below and fill in any answer boxes within your choice.
A. 25x^2(y-3)^2+35x(y-3)^2+49(y-3)^2=
B. The polynomial is prime.

Transcript text: Factor the trinomial completely. \[ 25 x^{2}(y-3)^{2}+35 x(y-3)^{2}+49(y-3)^{2} \] Select the correct choice below and fill in any answer boxes within your choice. A. $25 x^{2}(y-3)^{2}+35 x(y-3)^{2}+49(y-3)^{2}=$ $\square$ B. The polynomial is prime.

Solution

Solution Steps

Step 1: Identify the Common Factor

The given trinomial is: \[ 25 x^{2}(y-3)^{2}+35 x(y-3)^{2}+49(y-3)^{2} \] Notice that $(y-3)^2$ is a common factor in all three terms.

Step 2: Factor Out the Common Factor

Factor out $(y-3)^2$ from each term: \[ (y-3)^2 \left(25x^2 + 35x + 49\right) \]

Step 3: Check if the Remaining Trinomial Can Be Factored

Now, examine the trinomial inside the parentheses: \[ 25x^2 + 35x + 49 \] Check if this trinomial can be factored further. The discriminant $D = b^2 - 4ac$ is: \[ D = 35^2 - 4 \cdot 25 \cdot 49 = 1225 - 4900 = -3675 \] Since the discriminant is negative, the trinomial $25x^2 + 35x + 49$ cannot be factored further over the real numbers.

Step 4: Conclude the Factorization

Since the trinomial inside the parentheses cannot be factored further, the complete factorization of the original expression is: \[ (y-3)^2 \left(25x^2 + 35x + 49\right) \]

Step 5: Select the Correct Choice

The correct choice is: A. $25 x^{2}(y-3)^{2}+35 x(y-3)^{2}+49(y-3)^{2}=$ $(y-3)^2 (25x^2 + 35x + 49)$