What is the factored form of $a^2-121$ ?
The given expression is \(a^2 - 121\). We recognize this as a difference of squares, which follows the algebraic identity \(x^2 - y^2 = (x - y)(x + y)\). In our expression, we can set \(x = a\) and observe that \(121\) is a perfect square, as \(11^2 = 121\). Therefore, we can write the expression as \(a^2 - 11^2\).
Using the difference of squares identity, we substitute \(x = a\) and \(y = 11\) into the formula \((x - y)(x + y)\). This gives us \((a - 11)(a + 11)\). Comparing this result with the provided options:
We can clearly see that option c matches our calculated result.
Factoring expressions like \(a^2 - 121\) is useful in various real-life scenarios, such as simplifying algebraic models in engineering, physics, or finance. For example, if you are designing a structural component and the surface area is defined by the difference of two square areas \(A = L^2 - W^2\), factoring it into \((L-W)(L+W)\) can help you quickly calculate dimensions or analyze the material's properties under different loads. This ability to break down complex expressions into simpler, multiplicative factors is a fundamental skill that bridges the gap between theoretical math and practical problem-solving.