First PrinciplesStart anywhere. Prove it, then move on.

Rung 11

Polynomials and roots

Curves whose shape you can predict before you plot a single point.

Best attempted after 9. Exponents and their inverse. Nothing stops you trying this now — the gate will tell you if you were right.

The gate

Given a polynomial in factored form you have not seen before, sketch it without plotting points: mark every root, state the behaviour far left and far right, and say how many turns it can have. Then check against a plotter and account for every difference.

Nobody checks this but you. Do it honestly and the rungs above hold; do it loosely and they will not, somewhere further up where the cause is much harder to find.

A line was rung 6: one slope, no surprises. A polynomial bends, and how it bends is not arbitrary — the algebra tells you the shape before you draw anything.

Factored form is where this becomes obvious. y = (x - 1)(x + 2)(x - 4) announces its roots out loud: the product is zero exactly when one factor is zero.

Why this is on the ladder

Two reasons. First, reading structure off a formula — rather than evaluating it at a hundred points — is the skill that makes everything above rung 19 tractable. Second, roots. Enormous amounts of what follows reduce to where does this hit zero?, and at rung 41 the whole business of training a model is a search for a place where a derivative is zero.

Do this

Take y = (x - 1)(x + 2)(x - 4) and, before plotting:

  • Mark the roots. Three factors, three places it crosses.
  • Decide the far-right behaviour: for huge positive x every factor is positive, so the product shoots up. For huge negative x, all three are negative, so the product dives.
  • Count the turns. Degree three, so at most two.

Sketch it. Now check with a plotter, and if anything surprised you, work out why before moving on.

Then try y = (x - 1)^2 (x + 3). The repeated factor behaves differently at its root — it touches rather than crosses. Explain that from the algebra, not from the picture.

Where people get stuck

Sign errors: (x + 2) has its root at x = -2, not +2. Say "what makes this factor zero?" rather than reading the number off.

The repeated-root case catches almost everyone. Near x = 1, (x - 1)^2 is positive on both sides — it never changes sign — so the curve bounces. That is the algebra explaining the geometry, which is the entire point of the rung.

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