195. Expanding Binomial Products

Learning Intentions

• Identify terms in two binomial factors.
• Expand binomial products Use systematic multiplication.
• Simplify expanded expressions by collecting like terms.

Pre-requisite Summary

• Know that a term is a number, pronumeral or product of numbers and pronumerals separated by or signs.

• Know that a binomial has two terms, such as or .

• Know that expanding means removing brackets by multiplication.

• Know that each term in one bracket must be multiplied by each term in the other bracket.

• Know that like terms contain the same pronumerals with the same powers.

• Know that expanded binomial products often contain like terms that need to be collected.

Worked Examples

Worked Example 1

Identify the two terms in each binomial factor.

a)

b)

c)

Worked Example 2

Identify the four products needed to expand each binomial product.

a)

b)

c)

Worked Example 3

Expand each binomial product using systematic multiplication.

a)

b)

c)

Worked Example 4

Expand each binomial product using systematic multiplication.

a)

b)

c)

Worked Example 5

Expand each binomial product.

a)

b)

c)

Worked Example 6

Expand each binomial product.

a)

b)

c)

Worked Example 7

Expand and simplify each expression by collecting like terms.

a)

b)

c)

Worked Example 8

Expand and simplify each expression by collecting like terms.

a)

b)

c)

Problems

Problem 1

Identify the two terms in each binomial factor.

a)

b)

c)

Problem 2

Identify the four products needed to expand each binomial product.

a)

b)

c)

Problem 3

Expand each binomial product using systematic multiplication.

a)

b)

c)

Problem 4

Expand each binomial product using systematic multiplication.

a)

b)

c)

Problem 5

Expand each binomial product.

a)

b)

c)

Problem 6

Expand each binomial product.

a)

b)

c)

Problem 7

Expand and simplify each expression by collecting like terms.

a)

b)

c)

Problem 8

Expand and simplify each expression by collecting like terms.

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1

Identify the two terms in each binomial factor.

a)

b)

c)

d)

Exercise 2

Identify the four products needed to expand each binomial product.

a)

b)

c)

d)

Exercise 3

Expand each binomial product using systematic multiplication.

a)

b)

c)

d)

Exercise 4

Expand each binomial product using systematic multiplication.

a)

b)

c)

d)

Exercise 5

Expand each binomial product.

a)

b)

c)

d)

Exercise 6

Expand each binomial product.

a)

b)

c)

d)

Exercise 7

Expand and simplify each expression by collecting like terms.

a)

b)

c)

d)

Exercise 8

Expand and simplify each expression by collecting like terms.

a)

b)

c)

d)

Exercise 9

Copy and complete each expansion.

a)

b)

c)

Exercise 10

Expand and simplify each expression.

a)

b)

c)

d)

Reasoning

Exercise 11

Explain why does not equal .

Exercise 12

A student expands:

Explain the mistake and write the correct expansion.

Exercise 13

A student expands:

Explain the sign error and write the correct expansion.

Exercise 14

Explain why the two middle terms in can be collected.

Exercise 15

Decide whether each expansion is correct. Justify your answer.

a)

b)

c)

Exercise 16

Use systematic multiplication to explain why:

Problem-solving

Exercise 17

A rectangle has length and width .

a) Write an expression for the area.

b) Expand the expression.

c) Simplify the expression by collecting like terms.

Exercise 18

A rectangle has length and width .

a) Write an expression for the area.

b) Expand the expression using systematic multiplication.

c) Simplify the expression.

Exercise 19

A student expands the area of a rectangle with side lengths and as:

a) Identify the incorrect term.

b) Correct the expansion.

c) Explain how signs should be handled when multiplying the terms.

Exercise 20

Create your own binomial product that expands to an expression with three terms after like terms are collected.

Your response must include:

• two binomial factors
• the four products from systematic multiplication
• the simplified expanded expression
• one sentence explaining which terms were collected

Potential Misunderstandings

• Students may identify only the pronumeral terms and ignore constant terms in binomials.

• Students may miss negative signs when identifying terms, such as treating as having terms and instead of and .

• Students may not Recognise that both brackets are factors in a product.

• Students may multiply only the first terms and last terms, such as writing .

• Students may forget that every term in the first bracket must multiply every term in the second bracket.

• Students may make sign errors when multiplying positive and negative terms.

• Students may combine unlike terms, such as writing .

• Students may forget to collect the two middle terms after expanding.

• Students may write the correct four products but make arithmetic errors when simplifying the coefficient of the term.

Next: 196. Expanding Special Binomial Products