Factor polynomial expressions completely

Factoring completely means rewriting a polynomial expression as a product of a greatest common factor and irreducible factors by identifying structure such as common factors, grouping, quadratic patterns, and special products including difference of squares and sums or differences of cubes. The factorization preserves equivalence and reveals zeros and multiplicities of the associated polynomial function; factors are taken as far as possible over the specified number system, without extending to complex-number factorization or abstract generalizations.

Detailed Explanation: Factor polynomial expressions completely

To factor a polynomial completely, keep looking for factors until none of the factors can be factored further over the integers. A good first step is always to identify the greatest common factor (GCF).

Example

Factor completely:

6x^4-24x^2

Step 1: Find the GCF.

The GCF of $6x^4$ and $24x^2$ is $6x^2$ . Factor it out:

6x^4-24x^2=6x^2(x^2-4)

Step 2: Look for a special product.

The expression inside the parentheses is a difference of squares:

x^2-4=x^2-2^2

Use the pattern

a^2-b^2=(a-b)(a+b)

So,

x^2-4=(x-2)(x+2)

Step 3: Write the complete factorization.

\boxed{6x^2(x-2)(x+2)}

This is completely factored because $6$ is a constant, $x^2$ cannot be factored further over the integers, and neither $x-2$ nor $x+2$ can be factored further.

To check, multiply:

6x^2(x-2)(x+2)=6x^2(x^2-4)=6x^4-24x^2

which matches the original expression.

Learn by doing: Factor polynomial expressions completely

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Factor Polynomials (Order 3) - By Grouping to Order 1 Factors, Coefficient 1

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