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Factor polynomial expressions completely

Factoring polynomial expressions completely means rewriting a polynomial as a product of factors by identifying and removing a greatest common factor, then applying structures such as difference of squares, perfect-square trinomials, and quadratic trinomials with suitable integer coefficients. The factored form is equivalent to the original by the distributive property and supports simplifying rational expressions and solving polynomial equations; this scope excludes advanced factorization of high-degree polynomials and methods over complex or other abstract number systems.

Detailed Explanation: Factor polynomial expressions completely

To factor a polynomial completely, rewrite it as a product of factors. Use this order:

  1. Find the greatest common factor (GCF).
  2. Look at what remains for a familiar pattern, such as a difference of squares.
  3. Check that none of the factors can be factored further.

Example

Factor completely:

6x3−24x6x^3-24x

Step 1: Find the GCF.

The GCF of 6x36x^3 and 24x24x is 6x6x. Factor it out:

6x3−24x=6x(x2−4)6x^3-24x=6x(x^2-4)

You can check this by distributing:

6x(x2)−6x(4)=6x3−24x6x(x^2)-6x(4)=6x^3-24x

Step 2: Factor the expression inside the parentheses.

The expression x2−4x^2-4 is a difference of squares because

x2−4=x2−22x^2-4=x^2-2^2

Use the pattern

a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b)

So,

x2−4=(x−2)(x+2)x^2-4=(x-2)(x+2)

Step 3: Write the complete factorization.

6x(x−2)(x+2)\boxed{6x(x-2)(x+2)}

This is completely factored because 66 is a constant, xx is a single variable factor, and neither x−2x-2 nor x+2x+2 can be factored further using integer factors.

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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