Look Again

Factoring completely — because correct and finished are not the same thing.

MATHEMATICS 8 MATATAG · Key Stage 3
CG Quarter 1 · competency 8
Term 1 · Week 7

The routine — in this order, every time

1Common factor? Take it out first.
2Two terms? Try difference of two squares.
3Three terms? Two numbers: product and sum.
4LOOK AGAIN — can any bracket go further?
5Check — expand it back.

All three are correct. One is finished.

2x² − 8
2(x² − 4)
not finished
(2x − 4)(x + 2)
not finished
2(x − 2)(x + 2)
complete

Every one expands back to 2x² − 8. The first leaves x² − 4 factorable; the second leaves a 2 inside 2x − 4. Nothing here is wrong — two of them are unfinished.

The three types you must know

TypePatternExample
Common monomial factor ab + ac = a(b + c) 6x² + 15x = 3x(2x + 5)
Difference of two squares a² − b² = (a − b)(a + b) x² − 64 = (x − 8)(x + 8)
Quadratic trinomial product and sum x² + 9x + 14 = (x + 2)(x + 7)
Perfect square trinomial a² + 2ab + b² = (a + b)² x² − 12x + 36 = (x − 6)²

A perfect square trinomial is not a fourth method — it is a trinomial whose two factors happen to be the same. Spotting it saves time; missing it costs none.

The two-steppers — where marks are lost

2x² − 72→ 2(x² − 36)→ 2(x − 6)(x + 6) CMF then DOTS
3x³ − 3x→ 3x(x² − 1)→ 3x(x − 1)(x + 1) CMF then DOTS
2x² + 14x + 20→ 2(x² + 7x + 10)→ 2(x + 2)(x + 5) CMF then trinomial

The amber middle column is correct. It is also where most answers stop.

These do not factor — and stopping is the right answer

x² + 4 x² + 25 x² + 2x + 3 x² + 5x + 9

A sum of two squares never factors. Nor does every trinomial: for x² + 2x + 3 you would need two numbers multiplying to 3 and adding to 2, and there are none.
“It stops here” is a complete answer.

Common factor first. Then look again.
MATATAG Mathematics CG (DepEd, August 2023), p. 56
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