Factor completely, or state that the polynomial is prime. 3x^4-12x^2

Verified step by step guidance

1. Step 1: Identify the greatest common factor (GCF) of the terms in the polynomial. The terms are 3x⁴ and -12x². The GCF is 3x².
2. Step 2: Factor out the GCF, 3x², from each term. This gives 3x²(x² - 4).
3. Step 3: Observe the remaining factor, x² - 4. Notice that it is a difference of squares, which can be factored further using the formula a² - b² = (a - b)(a + b).
4. Step 4: Apply the difference of squares formula to x² - 4. This results in (x - 2)(x + 2).
5. Step 5: Combine all the factors. The completely factored form of the polynomial is 3x²(x - 2)(x + 2).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process is essential for simplifying expressions and solving equations. Common techniques include factoring out the greatest common factor (GCF), using special products like the difference of squares, and applying methods such as grouping or the quadratic formula for higher-degree polynomials.

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that divides two or more numbers or terms without leaving a remainder. In polynomial expressions, identifying the GCF allows for simplification by factoring it out, which can make the remaining polynomial easier to work with. For example, in the polynomial 3x^4 - 12x^2, the GCF is 3x^2.

Prime Polynomials

A polynomial is considered prime if it cannot be factored into the product of two non-constant polynomials with real coefficients. Recognizing prime polynomials is crucial in algebra, as it indicates that the polynomial cannot be simplified further. In the context of the given polynomial, if after attempting to factor it, no further factorization is possible, it is classified as prime.