197. Factorising Monic Quadratics

Learning Intentions

• Identify monic quadratic expressions.
• Factorise monic quadratics by finding factor pairs.
• Check factorisation by expanding the binomial factors.

Pre-requisite Summary

• Know that a quadratic expression includes a squared variable term, such as .
• Know that a monic quadratic has coefficient on the term, such as .
• Know that factorising reverses expanding.
• Know that factor pairs are two numbers that multiply to give a product.
• Know how to expand binomial products such as .
• Know how to collect like terms after expanding.

Worked Examples

Worked Example 1

Identify whether each expression is a monic quadratic.

a)

b)

c)

Worked Example 2

For each monic quadratic, state the value of and in the form .

a)

b)

c)

Worked Example 3

Find the factor pair needed to factorise each expression.

a)

b)

c)

Worked Example 4

Factorise each monic quadratic.

a)

b)

c)

Worked Example 5

Factorise each monic quadratic.

a)

b)

c)

Worked Example 6

Factorise each monic quadratic.

a)

b)

c)

Worked Example 7

Check each factorisation by expanding the binomial factors.

a)

b)

c)

Problems

Problem 1

Identify whether each expression is a monic quadratic.

a)

b)

c)

Problem 2

For each monic quadratic, state the value of and in the form .

a)

b)

c)

Problem 3

Find the factor pair needed to factorise each expression.

a)

b)

c)

Problem 4

Factorise each monic quadratic.

a)

b)

c)

Problem 5

Factorise each monic quadratic.

a)

b)

c)

Problem 6

Factorise each monic quadratic.

a)

b)

c)

Problem 7

Check each factorisation by expanding the binomial factors.

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1

Identify whether each expression is a monic quadratic.

a)

b)

c)

d)

Exercise 2

For each monic quadratic, state the value of and in the form .

a)

b)

c)

d)

Exercise 3

Find the factor pair needed to factorise each expression.

a)

b)

c)

d)

Exercise 4

Find the factor pair needed to factorise each expression.

a)

b)

c)

d)

Exercise 5

Factorise each monic quadratic.

a)

b)

c)

d)

Exercise 6

Factorise each monic quadratic.

a)

b)

c)

d)

Exercise 7

Factorise each monic quadratic.

a)

b)

c)

d)

Exercise 8

Factorise each monic quadratic.

a)

b)

c)

d)

Exercise 9

Check each factorisation by expanding the binomial factors.

a)

b)

c)

d)

Exercise 10

Copy and complete each factorisation.

a)

b)

c)

d)

Reasoning

Exercise 11

Explain why is a monic quadratic, but is not.

Exercise 12

Explain why the factor pair for must multiply to and add to .

Exercise 13

A student factorises:

Explain why this is correct by expanding the binomial factors.

Exercise 14

A student factorises:

Explain the mistake and write the correct factorisation.

Exercise 15

A student factorises:

Explain the sign error and write the correct factorisation.

Exercise 16

Decide whether each factorisation is correct. Justify your answer.

a)

b)

c)

Problem-solving

Exercise 17

A rectangle has area .

a) Factorise the expression.

b) State possible expressions for the length and width.

c) Check by expanding the binomial factors.

Exercise 18

A rectangle has area .

a) Factorise the expression.

b) State possible expressions for the length and width.

c) Check by expanding the binomial factors.

Exercise 19

A student says that factorises to .

a) Check the factorisation by expanding.

b) Explain why the signs produce the middle term .

c) State whether the student is correct.

Exercise 20

Create your own monic quadratic that can be factorised.

Your response must include:

• the monic quadratic expression
• the factor pair used
• the factorised form
• a check by expanding the binomial factors

Potential Misunderstandings

• Students may think any quadratic is monic, even when the coefficient of is not .
• Students may confuse the coefficient of with the constant term.
• Students may not recognise that is monic because the coefficient of is understood to be .
• Students may find factor pairs that multiply to the constant term but forget to check that they add to the coefficient of .
• Students may ignore signs when factorising expressions such as .
• Students may choose the correct numbers but place the signs incorrectly.
• Students may think factorising changes the value of the expression rather than rewriting it in an equivalent form.
• Students may check a factorisation by only comparing the first and last terms.
• Students may forget to collect like terms after expanding the binomial factors.

Next: 198. Connecting Expansion and Factorisation