Every Pair of Terms, Every Sign

You'll learnto expand and factor polynomials, keeping every term and sign, then report the object asked.

Expanding and factoring are the same move run in two directions — one multiplies a product out, the other runs it backward. Watch where careful students still leak points: dropping a square's middle term, mishandling the difference of squares, a sign slip inside the factor pair, and handing back a factor when a root was asked. Keep the middle term, read the sign between the squares, hold every sign steady, then report the exact object the item names.

8 STEPS · 6 QUICK CHECKS · a square keeps its middle term · factor ≠ root · v4

expand · factor

Before you start

What you're looking at

An area model of (x + 4)² split into four tiles — x², two cross tiles reading 4x, and 16. The panel on the right names the four leaks this animation walks in turn, one per chapter.

The question

When you square (x + 4), where does the middle term come from — and why is it easy to lose?

Watch for

The two shaded cross tiles each read 4x; together they are the 8x middle term that a squared sum quietly drops.

What the symbols mean

The big square's side is x + 4; each tile is one of the four products, and the tile areas sum to the expansion.

The parts

1 · Expand, keep the middle term → 2 · Difference of squares → 3 · The factor pair → 4 · Zero-product roots — then the shared trap: a factor is not its root.

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