Integrated_Algebra // Mod_06

FACTORING

The art of deconstruction. Factoring is reverse multiplication—breaking a polynomial down into its component parts to reveal its roots.

01 // The Diamond Method

Set Parameters
Finding factors for:
x² + 5x + 6
Product (c)
6
5
Sum (b)
?
?
No integer factors found

Why this works

When multiplying binomials (x+m)(x+n), the result expands to:

x2+(m+n)x+(mn)x^2 + (m+n)x + (mn)

Therefore, to go backwards from standard form, we need to find two mystery numbers that multiply to the constant term and add to the linear coefficient.

02 // The Strategy Toolkit

Always check this first!

Greatest Common Factor

ab+ac=a(b+c)ab + ac = a(b + c)

The universal first step. Extracting the largest value that divides evenly into every term.

2 Terms (Subtraction)

Difference of Perfect Squares

a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b)

A binomial where two perfect squares are subtracted. The middle term cancels out, leaving a symmetrical result.

3 Terms (a = 1)

Simple Trinomials

x2+bx+cx^2 + bx + c

The classic puzzle. Find two numbers that multiply to 'c' and add up to 'b'.

3 Terms (a > 1)

Complex Trinomials

ax2+bx+cax^2 + bx + c

Requires the 'AC Method' or 'Slide and Divide'. Splitting the middle term to factor by grouping.

4 Terms

Factor by Grouping

ax+ay+bx+byax + ay + bx + by

Splitting the polynomial in half, factoring the GCF from each side, and finding the common binomial.

2 Terms (Cubes)

Sum/Diff of Cubes

a3±b3a^3 \pm b^3

A specific pattern for cubic binomials. Remember the mnemonic 'SOAP' (Same, Opposite, Always Positive).

Geometric Proofs

Factoring isn't just symbol manipulation; it describes physical space. The "Difference of Squares" can be proven by taking a square of side aa, cutting out a square of side bb, and rearranging the remaining area into a new rectangle!

a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b)

Deconstruction Mastered

You are ready to manipulate functions.

Next: Functions