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Completely Factored, Prime Polynomials, and Finding Roots

Tie it together: a clean strategy that always tries the GCF first, knowing when a polynomial is prime, recognizing a completely factored answer, and using the zero-product property to turn factors into the roots of an equation.

A factoring strategy, and what counts as done

  1. GCF first — always pull out the greatest common factor of every term.
  2. Count terms: two terms → try difference of squares or sum/difference of cubes; three terms → trinomial or perfect square; four terms → grouping.
  3. Factor each piece again until nothing splits further.
  4. Check by multiplying everything back out.

An answer is [[completely-factored|completely factored]] when no factor can be broken down any further over the numbers you are allowed to use. A polynomial that cannot be factored at all (other than the trivial 1 times itself) is a [[prime-polynomial|prime polynomial]] — the algebraic cousin of a prime number. Examples: x^2 + 1 and x^2 + x + 1 are prime over the reals.

The zero-product property

Here is the payoff factoring was building toward. The [[zero-product-property|zero-product property]] says: if a product equals zero, then at least one factor must be zero. This is true only for zero — knowing a product equals 12 tells you nothing about the individual factors. So to solve an equation, move everything to one side so it equals 0, factor, and set each factor equal to 0. Each factor hands you a root.

Solve x^2 + 7x + 12 = 0      (already equals 0)

Factor:        (x + 3)(x + 4) = 0
Zero-product:  x + 3 = 0   or   x + 4 = 0
Roots:         x = -3       or   x = -4

Watch out — set the equation to 0 FIRST:
Solve x^2 = 5x + 6
  x^2 - 5x - 6 = 0
  (x - 6)(x + 1) = 0
  x = 6  or  x = -1   (NOT from x^2 = 5x+6 directly)
[[factoring-to-solve|Factoring to solve]]: set to 0, factor, then each factor = 0.

Double roots and a sanity check

When a factor repeats, you get a [[double-root|double root]] — one root counted twice. From (x - 5)^2 = 0 the only solution is x = 5, but it is a double root. The number of factors (counting repeats) always matches the equation's degree, which is a quiet reassurance you have found them all.