Question: Factor the following polynomial. \[ 320s^2 - 245v^2 \] \[ 320s^2 - 245v^2 =…

Factor the following polynomial.

\[ 320s^2 - 245v^2 \]

\[ 320s^2 - 245v^2 = \boxed{\phantom{Factor completely.}} \] (Factor completely.)

Solution

To factor the polynomial \(320s^2 - 245v^2\), we first look for the greatest common factor of the coefficients. The greatest common factor of 320 and 245 is 5. Factor out the greatest common factor: \[ 320s^2 - 245v^2 = 5(64s^2 - 49v^2) \] Notice that \(64s^2 - 49v^2\) is a difference of squares: \((8s)^2 - (7v)^2\) Apply the difference of squares formula, \(a^2 - b^2 = (a-b)(a+b)\): \[ 5(64s^2 - 49v^2) = 5((8s - 7v)(8s + 7v)) \] So, the completely factored form is: \[ 5(8s - 7v)(8s + 7v) \]