Problem 32 - Introductory Algebra for College Students

Question

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state. $$12 x^{2}-4 x^{4}$$

Step-by-Step Solution

Answer

The factored form of the polynomial is \(4x^{2}(3 - x^{2})\).

Step 1: Identify the Common Factor

Firstly, we need to identify the common factor between the terms of polynomial. Looking at the equation \(12x^{2} - 4x^{4}\), we note that both terms share a common factor of \(4x^{2}\).

Step 2: Factor out the Common Factor

Next, we factor the polynomial by removing the common factor. This gives us \(4x^{2}(3 - x^{2})\).

Step 3: Check the Result

Finally, we need to double-check our result. We see that \(4x^{2}(3 - x^{2})\) indeed equals the original polynomial \(12x^{2} - 4x^{4}\), confirming the factorization is correct.