Remediation sheet · Mathematics 8

What Do They Share?

Term 1 · Week 7 · Factoring polynomials completely
NameDate
For the teacher. This is for a learner who stops at the first factor, or treats a plain number as not a factor. It is not a shortened copy of worksheet.html — it is a different route to the same competency, starting further back. Task 4 is the first item on worksheet.html, reached from numbers rather than from letters. Show each worked example first and let the learner copy the method, never the answer. If task 1 will not come: If task 1 will not come, stay on numbers all lesson. Factoring letters adds nothing new to the idea.

1Numbers only — targets the idea of a common factor, before any algebra

Say: “What goes into both?”

Worked example. 12 and 18 both divide by 6. 12 = 6 × 2 and 18 = 6 × 3.
  1. 15 and 20 share ______ 15 = ______ × ______ 20 = ______ × ______
  2. 14 and 21 share ______ 14 = ______ × ______ 21 = ______ × ______
  3. 24 and 36 share ______ (the biggest one)

2Take the number out — targets "a common factor of 2 does not count"

Say: “Write it outside. What is left goes inside.”

Worked example. 2x − 4 = 2(x − 2). Check: 2 × x = 2x and 2 × −2 = −4. ✓
  1. 3x + 9 = ______(______ + ______)
  2. 5y − 20 = ______(______ − ______)
  3. 4a + 6 = ______(______ + ______)

3Take the letter out — targets "the common factor must be a number"

Say: “Is there an x in every part?”

Worked example. x² + 3x = x(x + 3). Every term had an x, so the x comes out.
  1. y² − 5y = ______(______ − ______)
  2. a² + a = ______(______ + ______)
  3. 2n² + n = ______(______ + ______)

4Both, then check — targets stopping while a factor remains

Say: “Look again. Is anything still shared?”

Worked example. 5x² + 15x: take 5 → 5(x² + 3x). Look again — every part still has an x → 5x(x + 3). Expand to check. ✓
  1. 6y² + 9y = ______________ Check by expanding: ______________
  2. 4a² − 8a = ______________ Check by expanding: ______________

Answer key — not printed on the learner's copy

  1. 5 (5×3, 5×4) · 7 (7×2, 7×3) · 12
  2. 3(x + 3) · 5(y − 4) · 2(2a + 3)
  3. y(y − 5) · a(a + 1) · n(2n + 1)
  4. 3y(2y + 3) · 4a(a − 2) — both expand back to the original