Kenneth ChengMathematics · models · ideas
Search中文

PAPER · REASONING · PRACTICE

Factorisation: common factors and square patterns — Teacher reference

Choose a method, factor completely over integers and check by expanding.

40 minutes · Year 8, upper volume · text pp. 73–78 · foundation lessons 37–39

Textbook reference: Year 8, upper · Chapter 14: polynomial operations and factorisation · lessons 37–39

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the upper-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Student classroom pageArrange in My Classroom

  1. 4′Check prerequisites
  2. 5′Observe and predict
  3. 8′Worked example and guided attempt
  4. 15′Independent practice
  5. 5′Discuss and correct
  6. 3′Exit questions

Observe · explain

A question before practising

Match an expansion card to its factorised product. Which features distinguish a difference of squares from a complete square?

Organise the key idea

Factorisation ends as a product. Extract common factors first, then inspect each remaining factor for a square pattern.

Open the original chapterOpen the interactive inquiry

Use the lesson question here for paper discussion; the original activity opens separately.

Worked example1

Factorise completely over the integers. Verify by expansion.

2x3−18x2x^3-18x
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Guided attempt2

Factorise completely over the integers. Verify by expansion.

x2−6x+9x^2-6x+9
  1. 9=___²; −6x=−2·x·___.
  2. Therefore the product is (___−___)².
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Prerequisite3

Factorise completely over the integers. Verify by expansion.

6x2+9x6x^2+9x
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Prerequisite4

Factorise completely over the integers. Verify by expansion.

x2−16x^2-16
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Independent practice5

Factorise completely over the integers. Verify by expansion.

8x2−12x8x^2-12x
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Independent practice6

Factorise completely over the integers. Verify by expansion.

9x2−259x^2-25
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Independent practice7

Factorise completely over the integers. Verify by expansion.

x2+10x+25x^2+10x+25
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Independent practice8

Factorise completely over the integers. Verify by expansion.

3x2−123x^2-12
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Exit question9

Factorise completely over the integers. Verify by expansion.

x2−4x+4x^2-4x+4
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Exit question10

Factorise completely over the integers. Verify by expansion.

2x2+6x2x^2+6x
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Further practice11

Factorise completely over the integers. Verify by expansion.

y2−49y^2-49
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Further practice12

Factorise completely over the integers. Verify by expansion.

4x2+12x+94x^2+12x+9
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Further practice13

Factorise completely over the integers. Verify by expansion.

(x+1)2−9(x+1)^2-9
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Further practice14

Factorise completely over the integers. Verify by expansion.

a2b−ab2a^2b-ab^2
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Extension15

Factorise completely over the integers. Verify by expansion.

x4−16x^4-16
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Extension16

Factorise completely over the integers. Verify by expansion.

8x3−32x8x^3-32x
Check your work
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Preparation and assessment notes

Prerequisites

Polynomial multiplication, difference of squares and complete squares.

Look for in students’ work

Use the worked model and guided attempt selectively, then the eight core questions. Keep further practice and extensions optional. Ask for a reason and an original-equation or expansion check.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

All reference answers

Worked example1

Factorise completely over the integers. Verify by expansion.

2x3−18x2x^3-18x
Reference answer

2x(x-3)(x+3).

2x(x−3)(x+3)2x(x-3)(x+3)
  1. Extract 2x first; the remaining x²−9 is a difference of squares.
  2. 2x³-18x = 2x(x²-9).
  3. The complete product is 2x(x-3)(x+3).
  4. Expanding the product gives 2x³-18x, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Guided attempt2

Factorise completely over the integers. Verify by expansion.

x2−6x+9x^2-6x+9
  1. 9=___²; −6x=−2·x·___.
  2. Therefore the product is (___−___)².
Reference answer

(x-3)².

(x−3)2(x-3)^2
  1. Check the two end squares and the middle term ±2ab before using (a±b)².
  2. x²-6x+9 = x²-2×x×3+3².
  3. The complete product is (x-3)².
  4. Expanding the product gives x²-6x+9, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Prerequisite3

Factorise completely over the integers. Verify by expansion.

6x2+9x6x^2+9x
Reference answer

3x(2x+3).

3x(2x+3)3x(2x+3)
  1. Extract the common numerical factor and each common variable to its smallest exponent.
  2. 6x²+9x = 3x(2x+3).
  3. The complete product is 3x(2x+3).
  4. Expanding the product gives 6x²+9x, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Prerequisite4

Factorise completely over the integers. Verify by expansion.

x2−16x^2-16
Reference answer

(x-4)(x+4).

(x−4)(x+4)(x-4)(x+4)
  1. Recognise two squares with a subtraction, then write their sum times their difference.
  2. x²-16 = x²-4².
  3. The complete product is (x-4)(x+4).
  4. Expanding the product gives x²-16, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Independent practice5

Factorise completely over the integers. Verify by expansion.

8x2−12x8x^2-12x
Reference answer

4x(2x-3).

4x(2x−3)4x(2x-3)
  1. Extract the common numerical factor and each common variable to its smallest exponent.
  2. 8x²-12x = 4x(2x-3).
  3. The complete product is 4x(2x-3).
  4. Expanding the product gives 8x²-12x, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Independent practice6

Factorise completely over the integers. Verify by expansion.

9x2−259x^2-25
Reference answer

(3x-5)(3x+5).

(3x−5)(3x+5)(3x-5)(3x+5)
  1. Recognise two squares with a subtraction, then write their sum times their difference.
  2. 9x²-25 = (3x)²-5².
  3. The complete product is (3x-5)(3x+5).
  4. Expanding the product gives 9x²-25, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Independent practice7

Factorise completely over the integers. Verify by expansion.

x2+10x+25x^2+10x+25
Reference answer

(x+5)².

(x+5)2(x+5)^2
  1. Check the two end squares and the middle term ±2ab before using (a±b)².
  2. x²+10x+25 = x²+2×x×5+5².
  3. The complete product is (x+5)².
  4. Expanding the product gives x²+10x+25, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Independent practice8

Factorise completely over the integers. Verify by expansion.

3x2−123x^2-12
Reference answer

3(x-2)(x+2).

3(x−2)(x+2)3(x-2)(x+2)
  1. Extract 3, then factor x²−4 by difference of squares.
  2. 3x²-12 = 3(x²-4).
  3. The complete product is 3(x-2)(x+2).
  4. Expanding the product gives 3x²-12, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Exit question9

Factorise completely over the integers. Verify by expansion.

x2−4x+4x^2-4x+4
Reference answer

(x-2)².

(x−2)2(x-2)^2
  1. Check the two end squares and the middle term ±2ab before using (a±b)².
  2. x²-4x+4 = x²-2×x×2+2².
  3. The complete product is (x-2)².
  4. Expanding the product gives x²-4x+4, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Exit question10

Factorise completely over the integers. Verify by expansion.

2x2+6x2x^2+6x
Reference answer

2x(x+3).

2x(x+3)2x(x+3)
  1. Extract the common numerical factor and each common variable to its smallest exponent.
  2. 2x²+6x = 2x(x+3).
  3. The complete product is 2x(x+3).
  4. Expanding the product gives 2x²+6x, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Further practice11

Factorise completely over the integers. Verify by expansion.

y2−49y^2-49
Reference answer

(y-7)(y+7).

(y−7)(y+7)(y-7)(y+7)
  1. Recognise two squares with a subtraction, then write their sum times their difference.
  2. y²-49 = y²-7².
  3. The complete product is (y-7)(y+7).
  4. Expanding the product gives y²-49, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Further practice12

Factorise completely over the integers. Verify by expansion.

4x2+12x+94x^2+12x+9
Reference answer

(2x+3)².

(2x+3)2(2x+3)^2
  1. Check the two end squares and the middle term ±2ab before using (a±b)².
  2. 4x²+12x+9 = (2x)²+2×(2x)×3+3².
  3. The complete product is (2x+3)².
  4. Expanding the product gives 4x²+12x+9, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Further practice13

Factorise completely over the integers. Verify by expansion.

(x+1)2−9(x+1)^2-9
Reference answer

(x-2)(x+4).

(x−2)(x+4)(x-2)(x+4)
  1. Recognise two squares with a subtraction, then write their sum times their difference.
  2. (x+1)²-9 = ((x+1)-3)((x+1)+3).
  3. The complete product is (x-2)(x+4).
  4. Expanding the product gives (x+1)²-9, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Further practice14

Factorise completely over the integers. Verify by expansion.

a2b−ab2a^2b-ab^2
Reference answer

ab(a-b).

ab(a−b)ab(a-b)
  1. Extract the common numerical factor and each common variable to its smallest exponent.
  2. a²b-ab² = ab(a-b).
  3. The complete product is ab(a-b).
  4. Expanding the product gives a²b-ab², the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Extension15

Factorise completely over the integers. Verify by expansion.

x4−16x^4-16
Reference answer

(x-2)(x+2)(x²+4).

(x−2)(x+2)(x2+4)(x-2)(x+2)(x^2+4)
  1. Apply difference of squares twice. Over integer coefficients, x²+4 has no further linear factors.
  2. x⁴-16 = (x²-4)(x²+4).
  3. The complete product is (x-2)(x+2)(x²+4).
  4. Expanding the product gives x⁴-16, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Extension16

Factorise completely over the integers. Verify by expansion.

8x3−32x8x^3-32x
Reference answer

8x(x-2)(x+2).

8x(x−2)(x+2)8x(x-2)(x+2)
  1. Extract 8x, then finish the difference-of-squares factor.
  2. 8x³-32x = 8x(x²-4).
  3. The complete product is 8x(x-2)(x+2).
  4. Expanding the product gives 8x³-32x, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Textbook reference and authorship

Chapter-core practice after the mapped textbook lessons; these tasks do not replace every first-teaching period.

大湾区初中数学三维课堂（八年级上册）正文.pdf

Year 8, upper volume · text pp. 73–78 · foundation lessons 37–39

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Factorisation: common factors and square patterns

Year 8, upper volume · text pp. 73–78 · foundation lessons 37–39 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Factorise completely over the integers. Verify by expansion.

2x3−18x2x^3-18x
Guided attempt2

Factorise completely over the integers. Verify by expansion.

x2−6x+9x^2-6x+9
  1. 9=___²; −6x=−2·x·___.
  2. Therefore the product is (___−___)².
Prerequisite3

Factorise completely over the integers. Verify by expansion.

6x2+9x6x^2+9x
Prerequisite4

Factorise completely over the integers. Verify by expansion.

x2−16x^2-16
g8-factorisation · 2026-10-14-upper-core-v1 · Conditions and reasons matter; measurement is not proof.

Factorisation: common factors and square patterns

Year 8, upper volume · text pp. 73–78 · foundation lessons 37–39 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Factorise completely over the integers. Verify by expansion.

8x2−12x8x^2-12x
Independent practice6

Factorise completely over the integers. Verify by expansion.

9x2−259x^2-25
Independent practice7

Factorise completely over the integers. Verify by expansion.

x2+10x+25x^2+10x+25
Independent practice8

Factorise completely over the integers. Verify by expansion.

3x2−123x^2-12
Exit question9

Factorise completely over the integers. Verify by expansion.

x2−4x+4x^2-4x+4
Exit question10

Factorise completely over the integers. Verify by expansion.

2x2+6x2x^2+6x
g8-factorisation · 2026-10-14-upper-core-v1 · Conditions and reasons matter; measurement is not proof.

Factorisation: common factors and square patterns — Further practice

Year 8, upper volume · text pp. 73–78 · foundation lessons 37–39

Further practice11

Factorise completely over the integers. Verify by expansion.

y2−49y^2-49
Further practice12

Factorise completely over the integers. Verify by expansion.

4x2+12x+94x^2+12x+9
Further practice13

Factorise completely over the integers. Verify by expansion.

(x+1)2−9(x+1)^2-9
Further practice14

Factorise completely over the integers. Verify by expansion.

a2b−ab2a^2b-ab^2
Extension15

Factorise completely over the integers. Verify by expansion.

x4−16x^4-16
Extension16

Factorise completely over the integers. Verify by expansion.

8x3−32x8x^3-32x

Factorisation: common factors and square patterns — Teacher answers

Year 8, upper volume · text pp. 73–78 · foundation lessons 37–39

Use the worked model and guided attempt selectively, then the eight core questions. Keep further practice and extensions optional. Ask for a reason and an original-equation or expansion check.

Worked example1

Factorise completely over the integers. Verify by expansion.

2x3−18x2x^3-18x
Reference answer

2x(x-3)(x+3).

2x(x−3)(x+3)2x(x-3)(x+3)
  1. Extract 2x first; the remaining x²−9 is a difference of squares.
  2. 2x³-18x = 2x(x²-9).
  3. The complete product is 2x(x-3)(x+3).
  4. Expanding the product gives 2x³-18x, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Guided attempt2

Factorise completely over the integers. Verify by expansion.

x2−6x+9x^2-6x+9
  1. 9=___²; −6x=−2·x·___.
  2. Therefore the product is (___−___)².
Reference answer

(x-3)².

(x−3)2(x-3)^2
  1. Check the two end squares and the middle term ±2ab before using (a±b)².
  2. x²-6x+9 = x²-2×x×3+3².
  3. The complete product is (x-3)².
  4. Expanding the product gives x²-6x+9, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Prerequisite3

Factorise completely over the integers. Verify by expansion.

6x2+9x6x^2+9x
Reference answer

3x(2x+3).

3x(2x+3)3x(2x+3)
  1. Extract the common numerical factor and each common variable to its smallest exponent.
  2. 6x²+9x = 3x(2x+3).
  3. The complete product is 3x(2x+3).
  4. Expanding the product gives 6x²+9x, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Prerequisite4

Factorise completely over the integers. Verify by expansion.

x2−16x^2-16
Reference answer

(x-4)(x+4).

(x−4)(x+4)(x-4)(x+4)
  1. Recognise two squares with a subtraction, then write their sum times their difference.
  2. x²-16 = x²-4².
  3. The complete product is (x-4)(x+4).
  4. Expanding the product gives x²-16, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Independent practice5

Factorise completely over the integers. Verify by expansion.

8x2−12x8x^2-12x
Reference answer

4x(2x-3).

4x(2x−3)4x(2x-3)
  1. Extract the common numerical factor and each common variable to its smallest exponent.
  2. 8x²-12x = 4x(2x-3).
  3. The complete product is 4x(2x-3).
  4. Expanding the product gives 8x²-12x, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Independent practice6

Factorise completely over the integers. Verify by expansion.

9x2−259x^2-25
Reference answer

(3x-5)(3x+5).

(3x−5)(3x+5)(3x-5)(3x+5)
  1. Recognise two squares with a subtraction, then write their sum times their difference.
  2. 9x²-25 = (3x)²-5².
  3. The complete product is (3x-5)(3x+5).
  4. Expanding the product gives 9x²-25, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Independent practice7

Factorise completely over the integers. Verify by expansion.

x2+10x+25x^2+10x+25
Reference answer

(x+5)².

(x+5)2(x+5)^2
  1. Check the two end squares and the middle term ±2ab before using (a±b)².
  2. x²+10x+25 = x²+2×x×5+5².
  3. The complete product is (x+5)².
  4. Expanding the product gives x²+10x+25, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Independent practice8

Factorise completely over the integers. Verify by expansion.

3x2−123x^2-12
Reference answer

3(x-2)(x+2).

3(x−2)(x+2)3(x-2)(x+2)
  1. Extract 3, then factor x²−4 by difference of squares.
  2. 3x²-12 = 3(x²-4).
  3. The complete product is 3(x-2)(x+2).
  4. Expanding the product gives 3x²-12, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Exit question9

Factorise completely over the integers. Verify by expansion.

x2−4x+4x^2-4x+4
Reference answer

(x-2)².

(x−2)2(x-2)^2
  1. Check the two end squares and the middle term ±2ab before using (a±b)².
  2. x²-4x+4 = x²-2×x×2+2².
  3. The complete product is (x-2)².
  4. Expanding the product gives x²-4x+4, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Exit question10

Factorise completely over the integers. Verify by expansion.

2x2+6x2x^2+6x
Reference answer

2x(x+3).

2x(x+3)2x(x+3)
  1. Extract the common numerical factor and each common variable to its smallest exponent.
  2. 2x²+6x = 2x(x+3).
  3. The complete product is 2x(x+3).
  4. Expanding the product gives 2x²+6x, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Further practice11

Factorise completely over the integers. Verify by expansion.

y2−49y^2-49
Reference answer

(y-7)(y+7).

(y−7)(y+7)(y-7)(y+7)
  1. Recognise two squares with a subtraction, then write their sum times their difference.
  2. y²-49 = y²-7².
  3. The complete product is (y-7)(y+7).
  4. Expanding the product gives y²-49, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Further practice12

Factorise completely over the integers. Verify by expansion.

4x2+12x+94x^2+12x+9
Reference answer

(2x+3)².

(2x+3)2(2x+3)^2
  1. Check the two end squares and the middle term ±2ab before using (a±b)².
  2. 4x²+12x+9 = (2x)²+2×(2x)×3+3².
  3. The complete product is (2x+3)².
  4. Expanding the product gives 4x²+12x+9, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Further practice13

Factorise completely over the integers. Verify by expansion.

(x+1)2−9(x+1)^2-9
Reference answer

(x-2)(x+4).

(x−2)(x+4)(x-2)(x+4)
  1. Recognise two squares with a subtraction, then write their sum times their difference.
  2. (x+1)²-9 = ((x+1)-3)((x+1)+3).
  3. The complete product is (x-2)(x+4).
  4. Expanding the product gives (x+1)²-9, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Further practice14

Factorise completely over the integers. Verify by expansion.

a2b−ab2a^2b-ab^2
Reference answer

ab(a-b).

ab(a−b)ab(a-b)
  1. Extract the common numerical factor and each common variable to its smallest exponent.
  2. a²b-ab² = ab(a-b).
  3. The complete product is ab(a-b).
  4. Expanding the product gives a²b-ab², the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Extension15

Factorise completely over the integers. Verify by expansion.

x4−16x^4-16
Reference answer

(x-2)(x+2)(x²+4).

(x−2)(x+2)(x2+4)(x-2)(x+2)(x^2+4)
  1. Apply difference of squares twice. Over integer coefficients, x²+4 has no further linear factors.
  2. x⁴-16 = (x²-4)(x²+4).
  3. The complete product is (x-2)(x+2)(x²+4).
  4. Expanding the product gives x⁴-16, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.
Extension16

Factorise completely over the integers. Verify by expansion.

8x3−32x8x^3-32x
Reference answer

8x(x-2)(x+2).

8x(x−2)(x+2)8x(x-2)(x+2)
  1. Extract 8x, then finish the difference-of-squares factor.
  2. 8x³-32x = 8x(x²-4).
  3. The complete product is 8x(x-2)(x+2).
  4. Expanding the product gives 8x³-32x, the original expression.
  • Finish as a product of integer-coefficient factors.
  • Continue factoring when possible and verify the expansion.