To factor a polynomial means to write it a product of polynomials.
EXAMPLE: 4x2 − 15x + 9 = (4x − 3)(x − 3). The method, by which this
factoring was obtained, will be explained below (see method D).
TIP: Whenever you want to factor a polynomial, try the four methods ex-
plained below, in the order A, B, C, D (if applicable)
A: Pulling out obvious factors.
B: Recognize one of the Special Products. These are:
• Difference of Squares: A2 − B2 = (A + B)(A − B);
• Square of a Sum: A2 + 2AB + B22 = (A + B)2;
• Square of a Difference: A2 − 2AB + B2 = (A − B)2;
• Difference of Cubes: A3 − B3 = (A − B)(A2 + AB + B2);
• Sum of Cubes: A3 + B3 = (A + B)(A2 − AB + B2).
• to factor 4x^2 − 25, we use the Difference of
Squares, with A = 2x and
B = 5:
• to factor 9y^2 −30y +25, we use the Square of a
Difference, with A = 3y
and B = 5:
• to factor 125z^3 + 8, we use the Sum of Cubes, with A = 5z and B = 2:
1. Split into (smaller) groups, which all share a common factor.
2. Pull that factor out.
For instance, to factor P(x) = x^3 + x^2 − 2x − 2, we form
two groups (using
square brackets), we factor them (separately), and finally we pull the (x+1)
D: Split-Group Factoring of Quadratic Trinomials. This
polynomials of the form
The method (”the p & q Game”) involves the following
1. Find numbers p and q, such that p · q = a · c and p + q = b
2. Split the polynomial into two groups: P(x) = [ax^2 + px] + [qx + c].
3. Factor by grouping (see method C). The two groups will always have a
For instance, suppose we want to factor:
(The coefficients are: a = 4, b = −15, and c = 9.)
We start off by looking for numbers p and q, such that p ·
q = 4 · 9 = 36
and p + q = −15. One working combination is: p = −12 and q = −3.
Then we split and factor by grouping: