To factor a polynomial, you generally follow these steps depending on the type and complexity of the polynomial:
Basic Steps to Factor Polynomials
1. Factor out the Greatest Common Factor (GCF):
Look for the largest factor common to all terms and factor it out first. This
simplifies the polynomial and makes further factoring easier
. 2. Identify the type of polynomial:
- Binomial: May be a difference of squares, sum/difference of cubes, etc.
- Trinomial: Often quadratic, can be factored into two binomials.
- Polynomials with four or more terms: May require factoring by grouping
3. For trinomials of the form ax2+bx+cax^2+bx+cax2+bx+c:
- If a=1a=1a=1, find two numbers that add to bbb and multiply to ccc. Use these to write the factors as (x+m)(x+n)(x+m)(x+n)(x+m)(x+n)
- If a≠1a\neq 1a=1, use the "ac method": multiply aaa and ccc, find two numbers that multiply to acacac and add to bbb, then split the middle term and factor by grouping
4. Factoring by grouping (for four-term polynomials):
- Group terms in pairs.
- Factor out the GCF from each group.
- If the remaining binomials are the same, factor them out as a common binomial factor
5. Special cases:
- Difference of squares: a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b)a2−b2=(a−b)(a+b).
- Sum/difference of cubes: Use formulas like a3±b3=(a±b)(a2∓ab+b2)a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)a3±b3=(a±b)(a2∓ab+b2)
Example: Factoring x2+6x+8x^2+6x+8x2+6x+8
- Identify b=6b=6b=6, c=8c=8c=8.
- Find two numbers that add to 6 and multiply to 8: 2 and 4.
- Write factors: (x+2)(x+4)(x+2)(x+4)(x+2)(x+4)
Example: Factoring 2x2−4x+3x−62x^2-4x+3x-62x2−4x+3x−6 by grouping
- Group: (2x2−4x)+(3x−6)(2x^2-4x)+(3x-6)(2x2−4x)+(3x−6).
- Factor each: 2x(x−2)+3(x−2)2x(x-2)+3(x-2)2x(x−2)+3(x−2).
- Factor out common binomial: (2x+3)(x−2)(2x+3)(x-2)(2x+3)(x−2)
This process can be adapted depending on the polynomial's degree and terms. Factoring polynomials often involves trial and error, especially for complex expressions, but starting with the GCF and recognizing patterns is key
