Problem
Factor completely. \[ 25 x^{2}-70 x y+49 y^{2} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice, A. $25 x^{2}-70 x y+49 y^{2}=\square$ (Factor completely.) B. The polynomial is prime.
Solution
Step 1 ：The given expression is a quadratic in the form of \(ax^2 - 2abxy + b^2y^2\). This is a perfect square trinomial and can be factored as \((ax - by)^2\). Here, a = 5, b = 7, x = x, and y = y.
Step 2 ：So, the factored form of the given expression is \((5x - 7y)^2\).
Step 3 ：The expanded form of the expression \((5x - 7y)^2\) is \(25x^2 - 70xy + 49y^2\), which is the same as the given expression.
Step 4 ：Therefore, the factored form of the given expression is indeed \((5x - 7y)^2\).
Step 5 ：Final Answer: \(25 x^{2}-70 x y+49 y^{2}=\boxed{(5x - 7y)^2}\)
