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To effectively factor a polynomial, the first crucial step is identifying the common factors within the terms of the expression. In the polynomial \(12b^4 + 3b^2\), we need to find a factor that appears in both terms. This involves looking at both the coefficients (numbers) and the variables.

- **Identifying within Coefficients**: Look at the coefficients 12 and 3. The greatest common factor (GCF) here is 3. - **Identifying within Variables**: Consider the variables. The smallest degree of \(b\) that appears in both terms is \(b^2\).
Combining these, the common factor of the polynomial is \(3b^2\). Recognizing this shared factor simplifies the process of factorizations tremendously.

Every polynomial can potentially be expressed as a product of simpler polynomials, which is a fundamental concept in algebra. Let's see how to factor the given polynomial using the common factor found earlier:

1. **Factor Out the Common Factor**: Once the common factor \(3b^2\) is identified, you divide each term of the polynomial by this factor \(3b^2\). 2. **Simplify the Polynomial**: The division leads to: \[ \frac{12b^4}{3b^2} + \frac{3b^2}{3b^2} = 4b^2 + 1 \]
Now, write the original polynomial as a product:\[ 3b^2(4b^2 + 1) \]
Thus, \(3b^2(4b^2 + 1)\) is the factored form of \(12b^4 + 3b^2\). This method of extracting the common element streamlines polynomial operations later.

Verification is a critical step to ensure your factorization work is accurate. This involves expanding the factored expression to see if it matches the original polynomial. Let's verify our factorization:

- **Expand the Expression**: Take \(3b^2(4b^2 + 1)\) and distribute \(3b^2\) across each term inside the parenthesis: \[ 3b^2 \times 4b^2 + 3b^2 \times 1 = 12b^4 + 3b^2 \]
- **Compare with the Original**: The result of the expansion, \(12b^4 + 3b^2\), matches the original polynomial.
This confirmation ensures the factored expression accurately represents the polynomial. Verification is essential to catch any mistakes and gain confidence in the factorization process.