To factor an expression means to rewrite it as a product of simpler expressions or numbers that multiply to give the original expression.
How to Factor Numbers
- Find all pairs of whole numbers that multiply to the given number. These pairs are the factors of the number. For example, factors of 8 are 1, 2, 4, and 8 because 1×8=8, 2×4=8
How to Factor Algebraic Expressions
Step 1: Factor out the Greatest Common Factor (GCF)
- Identify the largest number and variable that divides all terms in the expression.
- Divide each term by the GCF and factor it out.
- Example: For 2x3+18x2+10x2x^3+18x^2+10x2x3+18x2+10x, the GCF is 2x2x2x, so factor out 2x2x2x to get 2x(x2+9x+5)2x(x^2+9x+5)2x(x2+9x+5)
Step 2: Factoring Trinomials (expressions with three terms)
- For trinomials of the form x2+bx+cx^2+bx+cx2+bx+c (where the coefficient of x2x^2x2 is 1):
- Find two numbers that multiply to ccc and add to bbb.
- Write the factors as (x+m)(x+n)(x+m)(x+n)(x+m)(x+n) where mmm and nnn are those two numbers.
- Example: x2+6x+8x^2+6x+8x2+6x+8 factors to (x+2)(x+4)(x+2)(x+4)(x+2)(x+4) because 2×4=8 and 2+4=6
- For trinomials where the coefficient of x2x^2x2 is not 1 (e.g., ax2+bx+cax^2+bx+cax2+bx+c):
- Multiply aaa and ccc.
- Find two numbers that multiply to a×ca\times ca×c and add to bbb.
- Rewrite the middle term using these two numbers.
- Factor by grouping.
- Example: 2x2−5x−32x^2-5x-32x2−5x−3 is factored by splitting −5x-5x−5x into −6x+x-6x+x−6x+x, then factoring by grouping
Step 3: Special Cases
- Difference of squares: Expressions like x2−16x^2-16x2−16 factor as (x−4)(x+4)(x-4)(x+4)(x−4)(x+4) because 16=4216=4^216=42.
- Sum or difference of cubes: Use specific formulas for factoring these expressions
Summary
- Always start by factoring out the GCF.
- For trinomials with leading coefficient 1, find two numbers that add to the middle term's coefficient and multiply to the constant term.
- For trinomials with leading coefficient not 1, use the "ac method" and factor by grouping.
- Recognize special patterns like difference of squares.
This approach allows you to factor most polynomial expressions efficiently
