PAPER · REASONING · PRACTICE

Powers and polynomial multiplication

Consolidate the three power laws and distribute monomial or polynomial products accurately.

40 minutes · Year 8, upper volume · text pp. 51–62 · foundation lessons 26–31

Textbook reference: Year 8, upper · Chapter 14: polynomial operations and factorisation · lessons 26–31

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

Compare x³x⁴, (x³)⁴ and x³+x⁴. Predict which can be combined by a power law, then justify.

Organise the key idea

Different structures require different rules: add exponents for equal-base products, multiply them for powers of powers, and distribute every polynomial term.

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)(x+4)

Multiply every term in one bracket by every term in the other, then collect like terms.
Write one intermediate line without skipping products or signs.
The simplified form is 2x²+5x-12; explain each term.

Multiply every term in one bracket by every term in the other, then collect like terms.
$(2x-3)(x+4) = 2x^2+8x-3x-12$ .
Collect like terms to obtain 2x²+5x-12.
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.

(2x^2y)^3

2³=___; $(x^2)^3=x^{\square}$ ; y³=___.
Multiply the three factors: ___.

Raise every factor to the power, including the coefficient and its sign.
Write one intermediate line without skipping products or signs.
The simplified form is 8x⁶y³; explain each term.

Raise every factor to the power, including the coefficient and its sign.
$(2x^2y)^3 = 2^3(x^2)^3y^3$ .
Collect like terms to obtain 8x⁶y³.
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

Simplify completely, showing the relevant rule.

x^3x^4

For the same base, add exponents in a product and multiply exponents for a power of a power.
Write one intermediate line without skipping products or signs.
The simplified form is x⁷; explain each term.

For the same base, add exponents in a product and multiply exponents for a power of a power.
$x^3x^4 = x^{3+4}$ .
Collect like terms to obtain x⁷.
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.

Prerequisite4

Simplify completely, showing the relevant rule.

(x^3)^4

For the same base, add exponents in a product and multiply exponents for a power of a power.
Write one intermediate line without skipping products or signs.
The simplified form is x¹²; explain each term.

For the same base, add exponents in a product and multiply exponents for a power of a power.
$(x^3)^4 = x^{3\cdot 4}$ .
Collect like terms to obtain x¹².
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.

(3xy^2)^2

Raise every factor to the power, including the coefficient and its sign.
Write one intermediate line without skipping products or signs.
The simplified form is 9x²y⁴; explain each term.

Raise every factor to the power, including the coefficient and its sign.
$(3xy^2)^2 = 3^2x^2(y^2)^2$ .
Collect like terms to obtain 9x²y⁴.
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.

(-2x^2y)(3xy^3)

Raise every factor to the power, including the coefficient and its sign.
Write one intermediate line without skipping products or signs.
The simplified form is -6x³y⁴; explain each term.

Raise every factor to the power, including the coefficient and its sign.
$(-2x^2y)(3xy^3) = (-2\cdot 3)x^{2+1}y^{1+3}$ .
Collect like terms to obtain -6x³y⁴.
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 completely, showing the relevant rule.

3x(2x-5)

Multiply every term in one bracket by every term in the other, then collect like terms.
Write one intermediate line without skipping products or signs.
The simplified form is 6x²-15x; explain each term.

Multiply every term in one bracket by every term in the other, then collect like terms.
$3x(2x-5) = 3x\cdot 2x-3x\cdot 5$ .
Collect like terms to obtain 6x²-15x.
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 completely, showing the relevant rule.

(x+2)(x-5)

Multiply every term in one bracket by every term in the other, then collect like terms.
Write one intermediate line without skipping products or signs.
The simplified form is x²-3x-10; explain each term.

Multiply every term in one bracket by every term in the other, then collect like terms.
$(x+2)(x-5) = x^2-5x+2x-10$ .
Collect like terms to obtain x²-3x-10.
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+x^2

These are added like terms, not multiplied equal-base powers. Add the coefficients only.
Write one intermediate line without skipping products or signs.
The simplified form is 2x²; explain each term.

These are added like terms, not multiplied equal-base powers. Add the coefficients only.
$x^2+x^2 = (1+1)x^2$ .
Collect like terms to obtain 2x².
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.

-2x(x-4)

Multiply every term in one bracket by every term in the other, then collect like terms.
Write one intermediate line without skipping products or signs.
The simplified form is -2x²+8x; explain each term.

Multiply every term in one bracket by every term in the other, then collect like terms.
$-2x(x-4) = -2x\cdot x+(-2x)\cdot (-4)$ .
Collect like terms to obtain -2x²+8x.
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.

(a+b)^2(a+b)^3

For the same base, add exponents in a product and multiply exponents for a power of a power.
Write one intermediate line without skipping products or signs.
The simplified form is (a+b)⁵; explain each term.

For the same base, add exponents in a product and multiply exponents for a power of a power.
$(a+b)^2(a+b)^3 = (a+b)^{2+3}$ .
Collect like terms to obtain (a+b)⁵.
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.

(y^2)^5

For the same base, add exponents in a product and multiply exponents for a power of a power.
Write one intermediate line without skipping products or signs.
The simplified form is y¹⁰; explain each term.

For the same base, add exponents in a product and multiply exponents for a power of a power.
$(y^2)^5 = y^{2\cdot 5}$ .
Collect like terms to obtain y¹⁰.
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 completely, showing the relevant rule.

(-3a^2b)^2

Raise every factor to the power, including the coefficient and its sign.
Write one intermediate line without skipping products or signs.
The simplified form is 9a⁴b²; explain each term.

Raise every factor to the power, including the coefficient and its sign.
$(-3a^2b)^2 = (-3)^2(a^2)^2b^2$ .
Collect like terms to obtain 9a⁴b².
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 completely, showing the relevant rule.

(2x+1)(3x-2)

Multiply every term in one bracket by every term in the other, then collect like terms.
Write one intermediate line without skipping products or signs.
The simplified form is 6x²-x-2; explain each term.

Multiply every term in one bracket by every term in the other, then collect like terms.
$(2x+1)(3x-2) = 6x^2-4x+3x-2$ .
Collect like terms to obtain 6x²-x-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.

Extension15

Simplify completely, showing the relevant rule.

(2x^2)(-3x)(4x^3)

Raise every factor to the power, including the coefficient and its sign.
Write one intermediate line without skipping products or signs.
The simplified form is -24x⁶; explain each term.

Raise every factor to the power, including the coefficient and its sign.
$(2x^2)(-3x)(4x^3) = (2\cdot (-3)\cdot 4)x^{2+1+3}$ .
Collect like terms to obtain -24x⁶.
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

A rectangle has positive side lengths x+3 and x+1 units, with x>0. Express its area as a simplified polynomial.

(x+3)(x+1)

Multiply every term in one bracket by every term in the other, then collect like terms.
Write one intermediate line without skipping products or signs.
The simplified form is x²+4x+3; explain each term.

Multiply every term in one bracket by every term in the other, then collect like terms.
$(x+3)(x+1) = x^2+x+3x+3$ .
Collect like terms to obtain x²+4x+3.
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

Positive integer powers, signed coefficients, like terms and distributivity.

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. 51–62 · foundation lessons 26–31

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

Powers and polynomial multiplication

Year 8, upper volume · text pp. 51–62 · foundation lessons 26–31 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Simplify completely, showing the relevant rule.

(2x-3)(x+4)

Guided attempt2

Simplify completely, showing the relevant rule.

(2x^2y)^3

2³=___; $(x^2)^3=x^{\square}$ ; y³=___.
Multiply the three factors: ___.

Prerequisite3

Simplify completely, showing the relevant rule.

x^3x^4

Prerequisite4

Simplify completely, showing the relevant rule.

(x^3)^4

Powers and polynomial multiplication

Year 8, upper volume · text pp. 51–62 · foundation lessons 26–31 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Simplify completely, showing the relevant rule.

(3xy^2)^2

Independent practice6

Simplify completely, showing the relevant rule.

(-2x^2y)(3xy^3)

Independent practice7

Simplify completely, showing the relevant rule.

3x(2x-5)

Independent practice8

Simplify completely, showing the relevant rule.

(x+2)(x-5)

Exit question9

Simplify completely, showing the relevant rule.

x^2+x^2

Exit question10

Simplify completely, showing the relevant rule.

-2x(x-4)

Powers and polynomial multiplication — Further practice

Year 8, upper volume · text pp. 51–62 · foundation lessons 26–31

Further practice11

Simplify completely, showing the relevant rule.

(a+b)^2(a+b)^3

Further practice12

Simplify completely, showing the relevant rule.

(y^2)^5

Further practice13

Simplify completely, showing the relevant rule.

(-3a^2b)^2

Further practice14

Simplify completely, showing the relevant rule.

(2x+1)(3x-2)

Extension15

Simplify completely, showing the relevant rule.

(2x^2)(-3x)(4x^3)

Extension16

A rectangle has positive side lengths x+3 and x+1 units, with x>0. Express its area as a simplified polynomial.

(x+3)(x+1)

Powers and polynomial multiplication — Teacher answers

Year 8, upper volume · text pp. 51–62 · foundation lessons 26–31

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)(x+4)

Reference answer

2x²+5x-12.

2x^2+5x-12

Multiply every term in one bracket by every term in the other, then collect like terms.
$(2x-3)(x+4) = 2x^2+8x-3x-12$ .
Collect like terms to obtain 2x²+5x-12.
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.

(2x^2y)^3

2³=___; $(x^2)^3=x^{\square}$ ; y³=___.
Multiply the three factors: ___.

Reference answer

8x⁶y³.

8x^6y^3

Raise every factor to the power, including the coefficient and its sign.
$(2x^2y)^3 = 2^3(x^2)^3y^3$ .
Collect like terms to obtain 8x⁶y³.
Check the signs, coefficients and exponents in every term.

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

Prerequisite3

Simplify completely, showing the relevant rule.

x^3x^4

Reference answer

x⁷.

x^7

For the same base, add exponents in a product and multiply exponents for a power of a power.
$x^3x^4 = x^{3+4}$ .
Collect like terms to obtain x⁷.
Check the signs, coefficients and exponents in every term.

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

Prerequisite4

Simplify completely, showing the relevant rule.

(x^3)^4

Reference answer

x¹².

x^{12}

For the same base, add exponents in a product and multiply exponents for a power of a power.
$(x^3)^4 = x^{3\cdot 4}$ .
Collect like terms to obtain x¹².
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.

(3xy^2)^2

Reference answer

9x²y⁴.

9x^2y^4

Raise every factor to the power, including the coefficient and its sign.
$(3xy^2)^2 = 3^2x^2(y^2)^2$ .
Collect like terms to obtain 9x²y⁴.
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.

(-2x^2y)(3xy^3)

Reference answer

-6x³y⁴.

-6x^3y^4

Raise every factor to the power, including the coefficient and its sign.
$(-2x^2y)(3xy^3) = (-2\cdot 3)x^{2+1}y^{1+3}$ .
Collect like terms to obtain -6x³y⁴.
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 completely, showing the relevant rule.

3x(2x-5)

Reference answer

6x²-15x.

6x^2-15x

Multiply every term in one bracket by every term in the other, then collect like terms.
$3x(2x-5) = 3x\cdot 2x-3x\cdot 5$ .
Collect like terms to obtain 6x²-15x.
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 completely, showing the relevant rule.

(x+2)(x-5)

Reference answer

x²-3x-10.

x^2-3x-10

Multiply every term in one bracket by every term in the other, then collect like terms.
$(x+2)(x-5) = x^2-5x+2x-10$ .
Collect like terms to obtain x²-3x-10.
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+x^2

Reference answer

2x².

2x^2

These are added like terms, not multiplied equal-base powers. Add the coefficients only.
$x^2+x^2 = (1+1)x^2$ .
Collect like terms to obtain 2x².
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.

-2x(x-4)

Reference answer

-2x²+8x.

-2x^2+8x

Multiply every term in one bracket by every term in the other, then collect like terms.
$-2x(x-4) = -2x\cdot x+(-2x)\cdot (-4)$ .
Collect like terms to obtain -2x²+8x.
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.

(a+b)^2(a+b)^3

Reference answer

(a+b)⁵.

(a+b)^5

For the same base, add exponents in a product and multiply exponents for a power of a power.
$(a+b)^2(a+b)^3 = (a+b)^{2+3}$ .
Collect like terms to obtain (a+b)⁵.
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.

(y^2)^5

Reference answer

y¹⁰.

y^{10}

For the same base, add exponents in a product and multiply exponents for a power of a power.
$(y^2)^5 = y^{2\cdot 5}$ .
Collect like terms to obtain y¹⁰.
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 completely, showing the relevant rule.

(-3a^2b)^2

Reference answer

9a⁴b².

9a^4b^2

Raise every factor to the power, including the coefficient and its sign.
$(-3a^2b)^2 = (-3)^2(a^2)^2b^2$ .
Collect like terms to obtain 9a⁴b².
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 completely, showing the relevant rule.

(2x+1)(3x-2)

Reference answer

6x²-x-2.

6x^2-x-2

Multiply every term in one bracket by every term in the other, then collect like terms.
$(2x+1)(3x-2) = 6x^2-4x+3x-2$ .
Collect like terms to obtain 6x²-x-2.
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.

(2x^2)(-3x)(4x^3)

Reference answer

-24x⁶.

-24x^6

Raise every factor to the power, including the coefficient and its sign.
$(2x^2)(-3x)(4x^3) = (2\cdot (-3)\cdot 4)x^{2+1+3}$ .
Collect like terms to obtain -24x⁶.
Check the signs, coefficients and exponents in every term.

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

Extension16

A rectangle has positive side lengths x+3 and x+1 units, with x>0. Express its area as a simplified polynomial.

(x+3)(x+1)

Reference answer

x²+4x+3.

x^2+4x+3

Multiply every term in one bracket by every term in the other, then collect like terms.
$(x+3)(x+1) = x^2+x+3x+3$ .
Collect like terms to obtain x²+4x+3.
Check the signs, coefficients and exponents in every term.

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