196. Expanding Special Binomial Products

Learning Intentions

• Recognise common binomial products such as squares and differences.
• Expand binomial squares accurately.
• Compare expanded forms with their original factorised forms.

Pre-requisite Summary

• Know that a binomial has two terms, such as or .
• Know that a binomial product contains two binomial factors, such as .
• Know that a binomial square has the form or .
• Know that means , not .
• Know that a difference of squares has the form .
• Know that expanded and factorised forms are equivalent when they give the same value for every allowed value of the variable.

Worked Examples

Worked Example 1

Identify whether each expression is a binomial square, a difference of squares product, or neither.

a)

b)

c)

Worked Example 2

Identify the special binomial product.

a)

b)

c)

Worked Example 3

Expand each binomial square.

a)

b)

c)

Worked Example 4

Expand each binomial square.

a)

b)

c)

Worked Example 5

Expand each difference of squares product.

a)

b)

c)

Worked Example 6

Compare each factorised form with its expanded form. Decide whether they are equivalent.

a) and

b) and

c) and

Worked Example 7

Find the missing term so the expressions are equivalent.

a)

b)

c)

Problems

Problem 1

Identify whether each expression is a binomial square, a difference of squares product, or neither.

a)

b)

c)

Problem 2

Identify the special binomial product.

a)

b)

c)

Problem 3

Expand each binomial square.

a)

b)

c)

Problem 4

Expand each binomial square.

a)

b)

c)

Problem 5

Expand each difference of squares product.

a)

b)

c)

Problem 6

Compare each factorised form with its expanded form. Decide whether they are equivalent.

a) and

b) and

c) and

Problem 7

Find the missing term so the expressions are equivalent.

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1

Identify whether each expression is a binomial square, a difference of squares product, or neither.

a)

b)

c)

d)

Exercise 2

Identify the special binomial product.

a)

b)

c)

d)

Exercise 3

Expand each binomial square.

a)

b)

c)

d)

Exercise 4

Expand each binomial square.

a)

b)

c)

d)

Exercise 5

Expand each difference of squares product.

a)

b)

c)

d)

Exercise 6

Expand each expression.

a)

b)

c)

d)

Exercise 7

Expand each difference of squares product.

a)

b)

c)

d)

Exercise 8

Compare each factorised form with its expanded form. Decide whether they are equivalent.

a) and

b) and

c) and

d) and

Exercise 9

Find the missing term so the expressions are equivalent.

a)

b)

c)

d)

Exercise 10

Match each factorised form to its expanded form.

a)

b)

c)

Expanded forms:

Reasoning

Exercise 11

Explain why is not equal to .

Exercise 12

A student expands:

Explain the mistake and write the correct expansion.

Exercise 13

Explain why has no term after it is expanded and simplified.

Exercise 14

A student says that and have the same expanded form because both contain and . Explain why this is incorrect.

Exercise 15

Decide whether each statement is true or false. Justify your answer.

a)

b)

c)

Exercise 16

Use systematic multiplication to explain why:

Problem-solving

Exercise 17

A square has side length .

a) Write an expression for the area.

b) Expand the expression.

c) Explain why the expanded expression has three terms.

Exercise 18

A square has side length .

a) Write an expression for the area.

b) Expand the expression.

c) Interpret the expanded expression as an area.

Exercise 19

A rectangle has length and width .

a) Write an expression for the area.

b) Expand the expression.

c) Explain why this is a difference of squares product.

Exercise 20

Create your own pair of equivalent expressions involving a special binomial product.

Your response must include:

• one factorised form
• one expanded form
• the name of the special binomial product
• one sentence justifying why the two forms are equivalent

Potential Misunderstandings

• Students may think because they square only the first and last terms.
• Students may forget that means .
• Students may not recognise that the middle term in a binomial square comes from two products.
• Students may expand as instead of .
• Students may write the constant term in as negative, even though is positive.
• Students may make sign errors when multiplying two negative terms.
• Students may confuse a binomial square with a difference of squares product.
• Students may not recognise that simplifies to because the middle terms cancel.
• Students may compare factorised and expanded forms by appearance only, rather than checking by expansion or substitution.

Next; 197. Factorising Monic Quadratics