Questions: Factor completely. Select "Prime" if the polynomial cannot be factored. 50 c^2-128 d^2 50 c^2-128 d^2= Prime

Transcript text: Factor completely. Select "Prime" if the polynomial cannot be factored. \[ \begin{array}{r} 50 c^{2}-128 d^{2} \\ 50 c^{2}-128 d^{2}=\square \end{array} \] Prime

Solution

Solution Steps

To factor the given binomial \(50c^2 - 128d^2\), we can follow these steps:

Identify Common Factors: Check if there is a greatest common factor (GCF) for the coefficients 50 and 128.
Difference of Squares: Recognize that the expression is a difference of squares, which can be factored using the identity \(a^2 - b^2 = (a - b)(a + b)\).

Step 1: Identify the Expression

We start with the expression: \[ 50c^2 - 128d^2 \]

Step 2: Factor Out the Greatest Common Factor

The greatest common factor (GCF) of the coefficients 50 and 128 is 2. We can factor this out: \[ 50c^2 - 128d^2 = 2(25c^2 - 64d^2) \]

Step 3: Apply the Difference of Squares Formula

Next, we recognize that \(25c^2 - 64d^2\) is a difference of squares, which can be factored using the identity \(a^2 - b^2 = (a - b)(a + b)\): \[ 25c^2 - 64d^2 = (5c)^2 - (8d)^2 = (5c - 8d)(5c + 8d) \]

Step 4: Combine the Factors

Putting it all together, we have: \[ 50c^2 - 128d^2 = 2(5c - 8d)(5c + 8d) \]