What Is Factorization?

Try this

The tree on the left: $60$ is first split into a product of two numbers, then split again. What do the leaves at the end have in common? What do you get if you multiply all the leaves together?
The tree on the right: $x^{4} - 5 x^{2} + 4$ is also split into a product of two polynomials, then split again. What is the degree of each leaf?
In both trees, there comes a point where nothing splits. An integer that cannot be split is a prime; a polynomial that cannot be split is an irreducible polynomial.
Read from the top down, it is factorization. Read from the bottom up, it is multiplication. They are the same thing in two directions.

Factorization: writing a polynomial as a product of polynomials, where no factor can be split any further.

looks like a sum x^{2} - 1 = looks like a product (x + 1) (x - 1)

Right to left is multiplying polynomials, which you already know how to do. Left to right is factorization: multiplication played backwards. That is why you can always check a factorization by multiplying back.

Predict first

Which of these two steps is a factorization? A: $x^{2} - 1 = (x + 1) (x - 1)$ B: $x^{2} + 2 x + 1 = x (x + 2) + 1$

Only AOnly BBoth A and BNeither

Choose an answer first; then you can see the result.

Why

Why keep going until nothing splits? Integers make it clear.

Two trees, one idea

Step 1 / 6

Integers first
$30$ is not prime, so it can be written as a product of two smaller numbers.

$30 = 5 \times 6$
Keep asking
All the way down
Polynomials work the same way
Ask all the way down
Side by side

What is factoring completely good for? Simplifying fractions, solving equations, deciding divisibility: they all rely on it. For example, to solve $x^{4} - 1 = 0$ , once you write it as $(x - 1) (x + 1) (x^{2} + 1) = 0$ you can see straight away that $x = 1$ or $x = - 1$ .

Remove one condition

The phrase “cannot be split” hides a condition: which numbers the coefficients are allowed to be. The earlier chapters of this volume use rational numbers only.

Break it

Counterexample lab: remove one condition

Conclusion: $x^{2} - 2$ cannot be split

Over the rationals, $x^{2} - 2$ is a leaf with a ✓. Tap it and see.

Practice

Exercise 1Basic

Write $84$ as a product of primes. How many prime factors are there? (Count repeats too.)

Exercise 2Basic

Factor $x^{2} - 9$ .

Exercise 3Advanced

Factor $x^{4} - 16$ completely.

Exercise 4Challenge

Given $x^{2} + m x - 15 = (x + 3) (x + n)$ , find $m$ .

Try this

Why

Two trees, one idea

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Stopping halfway

The result is not a product