# FACTORING POLYNOMIALS

To factor a polynomial means to write it a product of polynomials.

EXAMPLE: 4x^{2} − 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: A ^{2} − B^{2} = (A + B)(A − B);
• Square of a Sum: A^{2} + 2AB + B2^{2} = (A + B)^{2};
• Square of a Difference: A^{2} − 2AB + B^{2} = (A − B)^{2};
• Difference of Cubes: A^{3} − B^{3} = (A − B)(A^{2} + AB + B^{2});
• Sum of Cubes: A^{3} + B^{3} = (A + B)(A^{2} − AB + B^{2}).**

For instance,

• 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:

**C: Grouping
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)

factor out:

**D: Split-Group Factoring of Quadratic Trinomials. This
refers to
polynomials of the form**

**The method (”the p & q Game”) involves the following
steps:
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
common factor!**

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: