Three Answers,
All Correct

What “completely factored” means

Week 7 · Day 1 CG Quarter 1 · competency 8 45 minutes

Expand these — you did them last week

(x + 3)(x − 3)	= x² − 9
(x + 4)²	= x² + 8x + 16
2(x − 1)(x + 5)	= 2x² + 8x − 10

You can go forwards.
Today we go backwards — and backwards has a complication.

Factor

2x² − 8

Two minutes. Everyone writes something.

Three answers came back

A

2(x² − 4)

B

(2x − 4)(x + 2)

C

2(x − 2)(x + 2)

Which one is correct? Hands up for each.

A. 2(x² − 4)

2(x² − 4)

= 2x² − 8 ✔

B. (2x − 4)(x + 2)

(2x − 4)(x + 2)

= 2x² + 4x − 4x − 8

= 2x² − 8 ✔

C. 2(x − 2)(x + 2) = 2x² − 8 ✔

All three

Every one of them multiplies back to 2x² − 8. All three are correct. So why does the textbook accept only one?

Look inside each answer

Answer	Can anything in it factor further?
A 2(x² − 4)	Yes — x² − 4 = (x − 2)(x + 2)
B (2x − 4)(x + 2)	Yes — 2x − 4 = 2(x − 2)
C 2(x − 2)(x + 2)	No. Finished

Completely factored

means no factor can be factored any further.

3x² − 27 — watch what I do first

3x² − 27

= 3(x² − 9) common factor first

= 3(x − 3)(x + 3) look again

Same shape. One is finished, one is not.

complete

5x(x + 3)

nothing left to factor

not complete

x(2x − 18)

2x − 18 = 2(x − 9)
→ 2x(x − 9)

Your turn — complete or not? Finish the ones that are not.

Given	Complete?	Completely factored
2(x² − 4)	no	2(x − 2)(x + 2)
(2x − 4)(x + 2)	no	2(x − 2)(x + 2)
3(x² − 9)	no	3(x − 3)(x + 3)
5x(x + 3)	yes	5x(x + 3)
x(2x − 18)	no	2x(x − 9)

Ask every learner this, and only this

“Can anything in there be factored again? Show me where you looked.”

1 Common factor? → take it out FIRST

2 LOOK AGAIN → can any bracket factor further?

3 Check → expand it back

Common factor first. Then look again.

Exit ticket

A learner writes 3x² − 27 = 3(x² − 9)

Is this correct? YES / NO
Is it completely factored? YES / NO
If not, finish it.
One sentence: what is the difference between the two questions?

Quarter sheet. Name at the top. Question 1 is the one I read first.