Before you can factor anything, you must be fluent with the building blocks the parent note quietly assumes. This page defines every one of them — from what a letter like x even means — and shows the picture behind each.
Picture it: imagine a number line. The variable x is a slider that can sit anywhere on that line.
Why the topic needs it: factoring works on expressions built from these boxes. Without a symbol for "any number", we could never write a general rule like a2−b2=(a−b)(a+b) that works for every number at once.
x0=1 (zero copies — by convention, an empty product is 1)
Picture it:x2 is the area of a square whose side is x. x3 is the volume of a cube of side x. That is literally why we say "squared" and "cubed".
Why the topic needs it: every polynomial term is a number times a power of x. Pulling out a common factor means finding the lowest power present — that only makes sense once you can compare exponents.
Picture it: think of a polynomial as a shelf of labelled boxes. Each box is a term; the label tells you its coefficient and power.
Recall
In 4a2b3−8ab4+12a3b2, how many terms are there and what is the coefficient of the middle term?
Answer ::: Three terms; the middle term −8ab4 has coefficient −8 (the minus sign belongs to the coefficient).
Why the topic needs it: factoring is the act of rewriting a polynomial (a sum of terms) as a product. You must be able to name terms and coefficients to talk about the process at all.
Why the topic needs it: identities like difference of cubes (a3−b3=(a−b)(a2+ab+b2)) live or die by exactly which term is negative. Sign discipline is non-negotiable.
How to find gcd of numbers: list what divides each, take the biggest shared one.
gcd(6,9,15): 6's divisors are 1,2,3,6; 9's are 1,3,9; 15's are 1,3,5,15. The biggest number in all three lists is 3.
For variables: take the lowest power that appears in every term (a term can't share a power higher than the smallest one it has).
Why the topic needs it: "common factor extraction" is finding the GCF and pulling it out front. This is the first move in nearly every factoring problem.
Picture it: a rectangle of height a and width b+c. Its total area is a(b+c). Split it down the middle and it's two rectangles, areas ab and ac. Same area, two ways of writing it.
Why the topic needs it: see Distributive Property. It is the foundation the whole topic stands on — grouping, identities, and GCF extraction are all applications of this one rule.
FOIL is just distributive property applied to two brackets: First, Outer, Inner, Last.
(x+2)(x+3)=Fx⋅x+Ox⋅3+I2⋅x+L2⋅3=x2+5x+6
Why the topic needs it: every identity in the parent note (difference of squares, cubes, perfect squares) is proved by expanding the right side and checking it collapses to the left.
Picture it: just as 3 and 4 are factors of 12 (because 3×4=12), (x+2) and (x+3) are factors of x2+5x+6.
Why the topic needs it: grouping works by creating a common binomial factor. You cannot "spot the shared (x+3)" if the word binomial and the idea of factor aren't second nature.
Why the topic needs it: factoring is not busywork — it is the fastest road to solving equations. The parent note's "why factor?" section rests entirely on this idea.
