Question

factor completely.
256r^{2}-196

Explanation:

Step1: Recognize difference of squares

The expression \(256r^2 - 196\) is a difference of squares, since \(256r^2=(16r)^2\) and \(196 = 14^2\). The formula for factoring a difference of squares is \(a^2 - b^2=(a + b)(a - b)\).
Let \(a = 16r\) and \(b = 14\), so we can factor it as \((16r + 14)(16r - 14)\).

Step2: Factor out common factors

Now, we notice that both terms in each binomial have a common factor. For \(16r + 14\), the greatest common factor (GCF) of 16 and 14 is 2. Factoring out 2 from \(16r + 14\) gives \(2(8r + 7)\). For \(16r - 14\), the GCF of 16 and 14 is also 2. Factoring out 2 from \(16r - 14\) gives \(2(8r - 7)\).

Step3: Combine the factored forms

Now we multiply the factored forms together. We had \((16r + 14)(16r - 14)=2(8r + 7)\times2(8r - 7)\). Multiplying the 2s together gives \(4(8r + 7)(8r - 7)\).

Answer:

\(4(8r + 7)(8r - 7)\)