PAPER · REASONING · PRACTICE

Special products and polynomial division

Practise difference of squares, complete squares and every-term division while retaining nonzero divisors.

40 minutes · Year 8, upper volume · text pp. 63–72 · foundation lessons 32–36

Textbook reference: Year 8, upper · Chapter 14: polynomial operations and factorisation · lessons 32–36

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.

Teacher reference Arrange in My Classroom

Lesson routePhase

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

Observe · explain

A question before practising

Expand (x+2)² with multiplication before using the formula. Where does 4x come from?

Organise the key idea

A squared binomial includes its middle term. For same-base division subtract exponents; a⁰=1 requires a≠0.

Open the original chapter Open the interactive inquiry

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

Worked example1

Simplify completely, showing the relevant rule.

(2x-3)^2

Use (a±b)²=a²±2ab+b²; retain the middle term.
Write one intermediate line without skipping products or signs.
The simplified form is 4x²-12x+9; explain each term.

Use (a±b)²=a²±2ab+b²; retain the middle term.
$(2x-3)^2 = (2x)^2-2\cdot (2x)\cdot 3+3^2$ .
Collect like terms to obtain 4x²-12x+9.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Guided attempt2

Simplify completely, showing the relevant rule.

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

The matching terms are a=___ and b=___.
a²−b²=___−___=___.

Use (a+b)(a−b)=a²−b², identifying both terms first.
Write one intermediate line without skipping products or signs.
The simplified form is 9x²-4; explain each term.

Use (a+b)(a−b)=a²−b², identifying both terms first.
$(3x+2)(3x-2) = (3x)^2-2^2$ .
Collect like terms to obtain 9x²-4.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Prerequisite3

Evaluate (−3)², including the sign.

(-3)^2

Identify the known quantities and the required quantity.
Write the relevant equality before calculating.
Substitute carefully, retain signs and units, and check the result.

The negative sign is inside the brackets, so both factors are −3.
(−3)(−3)=9.

Check your work

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Prerequisite4

Simplify; retain the original divisor restriction x≠0.

\frac{x^5}{x^2}

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
Write one intermediate line without skipping products or signs.
The simplified form is x³; explain each term.

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
$\frac{x^5}{x^2} = x^{5-2}$ .
Collect like terms to obtain x³.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Independent practice5

Simplify completely, showing the relevant rule.

(x+5)(x-5)

Use (a+b)(a−b)=a²−b², identifying both terms first.
Write one intermediate line without skipping products or signs.
The simplified form is x²-25; explain each term.

Use (a+b)(a−b)=a²−b², identifying both terms first.
$(x+5)(x-5) = x^2-5^2$ .
Collect like terms to obtain x²-25.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Independent practice6

Simplify completely, showing the relevant rule.

(x+4)^2

Use (a±b)²=a²±2ab+b²; retain the middle term.
Write one intermediate line without skipping products or signs.
The simplified form is x²+8x+16; explain each term.

Use (a±b)²=a²±2ab+b²; retain the middle term.
$(x+4)^2 = x^2+2\cdot x\cdot 4+4^2$ .
Collect like terms to obtain x²+8x+16.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Independent practice7

Simplify; retain the original divisor restriction x≠0.

\frac{x^3}{x^3}

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
Write one intermediate line without skipping products or signs.
The simplified form is 1; explain each term.

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
$\frac{x^3}{x^3} = x^{3-3}=x^0$ .
Collect like terms to obtain 1.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Independent practice8

Simplify; retain the original divisor restriction x≠0.

\frac{6x^3-9x^2}{3x}

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
Write one intermediate line without skipping products or signs.
The simplified form is 2x²-3x; explain each term.

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
$\frac{6x^3-9x^2}{3x} = \frac{6x^3}{3x}-\frac{9x^2}{3x}$ .
Collect like terms to obtain 2x²-3x.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Exit question9

Simplify completely, showing the relevant rule.

(x-2)^2

Use (a±b)²=a²±2ab+b²; retain the middle term.
Write one intermediate line without skipping products or signs.
The simplified form is x²-4x+4; explain each term.

Use (a±b)²=a²±2ab+b²; retain the middle term.
$(x-2)^2 = x^2-2\cdot x\cdot 2+2^2$ .
Collect like terms to obtain x²-4x+4.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Exit question10

Simplify completely, showing the relevant rule.

\frac{4x^2+8x}{4}

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
Write one intermediate line without skipping products or signs.
The simplified form is x²+2x; explain each term.

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
$\frac{4x^2+8x}{4} = \frac{4x^2}{4}+\frac{8x}{4}$ .
Collect like terms to obtain x²+2x.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Further practice11

Simplify completely, showing the relevant rule.

(x+6)(x-6)

Use (a+b)(a−b)=a²−b², identifying both terms first.
Write one intermediate line without skipping products or signs.
The simplified form is x²-36; explain each term.

Use (a+b)(a−b)=a²−b², identifying both terms first.
$(x+6)(x-6) = x^2-6^2$ .
Collect like terms to obtain x²-36.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Further practice12

Simplify completely, showing the relevant rule.

(3x+1)^2

Use (a±b)²=a²±2ab+b²; retain the middle term.
Write one intermediate line without skipping products or signs.
The simplified form is 9x²+6x+1; explain each term.

Use (a±b)²=a²±2ab+b²; retain the middle term.
$(3x+1)^2 = (3x)^2+2\cdot (3x)\cdot 1+1^2$ .
Collect like terms to obtain 9x²+6x+1.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Further practice13

Simplify; retain the original divisor restriction x≠0.

\frac{x^8}{x^5}

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
Write one intermediate line without skipping products or signs.
The simplified form is x³; explain each term.

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
$\frac{x^8}{x^5} = x^{8-5}$ .
Collect like terms to obtain x³.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Further practice14

Simplify; retain the original divisor restriction x≠0, y≠0.

\frac{-12x^3y^2}{3xy}

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
Write one intermediate line without skipping products or signs.
The simplified form is -4x²y; explain each term.

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
$\frac{-12x^3y^2}{3xy} = (-12/3)x^{3-1}y^{2-1}$ .
Collect like terms to obtain -4x²y.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Extension15

Simplify completely, showing the relevant rule.

(x+3)^2-(x-3)^2

Use (a±b)²=a²±2ab+b²; retain the middle term.
Write one intermediate line without skipping products or signs.
The simplified form is 12x; explain each term.

Use (a±b)²=a²±2ab+b²; retain the middle term.
$(x+3)^2-(x-3)^2 = (x^2+6x+9)-(x^2-6x+9)$ .
Collect like terms to obtain 12x.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Extension16

Simplify completely, showing the relevant rule.

(2x-1)^2-(2x+1)(2x-1)

Use (a+b)(a−b)=a²−b², identifying both terms first.
Write one intermediate line without skipping products or signs.
The simplified form is -4x+2; explain each term.

Use (a+b)(a−b)=a²−b², identifying both terms first.
$(2x-1)^2-(2x+1)(2x-1) = (4x^2-4x+1)-(4x^2-1)$ .
Collect like terms to obtain -4x+2.
Check the signs, coefficients and exponents in every term.

Check your work

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Preparation and assessment notes

Prerequisites

Distribution, positive powers, coefficient arithmetic and like terms.

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.

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.

Print edition

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. 63–72 · foundation lessons 32–36

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

Special products and polynomial division

Year 8, upper volume · text pp. 63–72 · foundation lessons 32–36 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Simplify completely, showing the relevant rule.

(2x-3)^2

Guided attempt2

Simplify completely, showing the relevant rule.

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

The matching terms are a=___ and b=___.
a²−b²=___−___=___.

Prerequisite3

Evaluate (−3)², including the sign.

(-3)^2

Prerequisite4

Simplify; retain the original divisor restriction x≠0.

\frac{x^5}{x^2}

Special products and polynomial division

Year 8, upper volume · text pp. 63–72 · foundation lessons 32–36 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Simplify completely, showing the relevant rule.

(x+5)(x-5)

Independent practice6

Simplify completely, showing the relevant rule.

(x+4)^2

Independent practice7

Simplify; retain the original divisor restriction x≠0.

\frac{x^3}{x^3}

Independent practice8

Simplify; retain the original divisor restriction x≠0.

\frac{6x^3-9x^2}{3x}

Exit question9

Simplify completely, showing the relevant rule.

(x-2)^2

Exit question10

Simplify completely, showing the relevant rule.

\frac{4x^2+8x}{4}

Special products and polynomial division — Further practice

Year 8, upper volume · text pp. 63–72 · foundation lessons 32–36

Further practice11

Simplify completely, showing the relevant rule.

(x+6)(x-6)

Further practice12

Simplify completely, showing the relevant rule.

(3x+1)^2

Further practice13

Simplify; retain the original divisor restriction x≠0.

\frac{x^8}{x^5}

Further practice14

Simplify; retain the original divisor restriction x≠0, y≠0.

\frac{-12x^3y^2}{3xy}

Extension15

Simplify completely, showing the relevant rule.

(x+3)^2-(x-3)^2

Extension16

Simplify completely, showing the relevant rule.

(2x-1)^2-(2x+1)(2x-1)

Special products and polynomial division — Teacher answers

Year 8, upper volume · text pp. 63–72 · foundation lessons 32–36

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

Simplify completely, showing the relevant rule.

(2x-3)^2

Reference answer

4x²-12x+9.

4x^2-12x+9

Use (a±b)²=a²±2ab+b²; retain the middle term.
$(2x-3)^2 = (2x)^2-2\cdot (2x)\cdot 3+3^2$ .
Collect like terms to obtain 4x²-12x+9.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Guided attempt2

Simplify completely, showing the relevant rule.

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

The matching terms are a=___ and b=___.
a²−b²=___−___=___.

Reference answer

9x²-4.

9x^2-4

Use (a+b)(a−b)=a²−b², identifying both terms first.
$(3x+2)(3x-2) = (3x)^2-2^2$ .
Collect like terms to obtain 9x²-4.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Prerequisite3

Evaluate (−3)², including the sign.

(-3)^2

Reference answer

The negative sign is inside the brackets, so both factors are −3.
(−3)(−3)=9.

Use the stated relationship and show the calculation.
State the result with its unit where appropriate.

Prerequisite4

Simplify; retain the original divisor restriction x≠0.

\frac{x^5}{x^2}

Reference answer

x³; x≠0.

x^3

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
$\frac{x^5}{x^2} = x^{5-2}$ .
Collect like terms to obtain x³.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Independent practice5

Simplify completely, showing the relevant rule.

(x+5)(x-5)

Reference answer

x²-25.

x^2-25

Use (a+b)(a−b)=a²−b², identifying both terms first.
$(x+5)(x-5) = x^2-5^2$ .
Collect like terms to obtain x²-25.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Independent practice6

Simplify completely, showing the relevant rule.

(x+4)^2

Reference answer

x²+8x+16.

x^2+8x+16

Use (a±b)²=a²±2ab+b²; retain the middle term.
$(x+4)^2 = x^2+2\cdot x\cdot 4+4^2$ .
Collect like terms to obtain x²+8x+16.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Independent practice7

Simplify; retain the original divisor restriction x≠0.

\frac{x^3}{x^3}

Reference answer

1; x≠0.

1

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
$\frac{x^3}{x^3} = x^{3-3}=x^0$ .
Collect like terms to obtain 1.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Independent practice8

Simplify; retain the original divisor restriction x≠0.

\frac{6x^3-9x^2}{3x}

Reference answer

2x²-3x; x≠0.

2x^2-3x

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
$\frac{6x^3-9x^2}{3x} = \frac{6x^3}{3x}-\frac{9x^2}{3x}$ .
Collect like terms to obtain 2x²-3x.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Exit question9

Simplify completely, showing the relevant rule.

(x-2)^2

Reference answer

x²-4x+4.

x^2-4x+4

Use (a±b)²=a²±2ab+b²; retain the middle term.
$(x-2)^2 = x^2-2\cdot x\cdot 2+2^2$ .
Collect like terms to obtain x²-4x+4.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Exit question10

Simplify completely, showing the relevant rule.

\frac{4x^2+8x}{4}

Reference answer

x²+2x.

x^2+2x

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
$\frac{4x^2+8x}{4} = \frac{4x^2}{4}+\frac{8x}{4}$ .
Collect like terms to obtain x²+2x.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Further practice11

Simplify completely, showing the relevant rule.

(x+6)(x-6)

Reference answer

x²-36.

x^2-36

Use (a+b)(a−b)=a²−b², identifying both terms first.
$(x+6)(x-6) = x^2-6^2$ .
Collect like terms to obtain x²-36.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Further practice12

Simplify completely, showing the relevant rule.

(3x+1)^2

Reference answer

9x²+6x+1.

9x^2+6x+1

Use (a±b)²=a²±2ab+b²; retain the middle term.
$(3x+1)^2 = (3x)^2+2\cdot (3x)\cdot 1+1^2$ .
Collect like terms to obtain 9x²+6x+1.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Further practice13

Simplify; retain the original divisor restriction x≠0.

\frac{x^8}{x^5}

Reference answer

x³; x≠0.

x^3

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
$\frac{x^8}{x^5} = x^{8-5}$ .
Collect like terms to obtain x³.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Further practice14

Simplify; retain the original divisor restriction x≠0, y≠0.

\frac{-12x^3y^2}{3xy}

Reference answer

-4x²y; x≠0, y≠0.

-4x^2y

Divide coefficients and subtract same-base exponents; divide every polynomial term. The divisor must be nonzero.
$\frac{-12x^3y^2}{3xy} = (-12/3)x^{3-1}y^{2-1}$ .
Collect like terms to obtain -4x²y.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Extension15

Simplify completely, showing the relevant rule.

(x+3)^2-(x-3)^2

Reference answer

12x.

12x

Use (a±b)²=a²±2ab+b²; retain the middle term.
$(x+3)^2-(x-3)^2 = (x^2+6x+9)-(x^2-6x+9)$ .
Collect like terms to obtain 12x.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.

Extension16

Simplify completely, showing the relevant rule.

(2x-1)^2-(2x+1)(2x-1)

Reference answer

-4x+2.

-4x+2

Use (a+b)(a−b)=a²−b², identifying both terms first.
$(2x-1)^2-(2x+1)(2x-1) = (4x^2-4x+1)-(4x^2-1)$ .
Collect like terms to obtain -4x+2.
Check the signs, coefficients and exponents in every term.

Show the rule and a complete intermediate line.
Give an algebraically equivalent simplified expression.