Expanding Binomial Products
Match the Expansion
Draw a line from each binomial product to its expanded form.
Identify the Correct Expansion
Circle the correct expansion for each binomial product.
(x + 3)(x + 5)
(x - 4)(x + 2)
(x - 6)(x - 1)
(2x + 1)(x + 3)
Expand Using FOIL
Expand each product using FOIL (First, Outer, Inner, Last). Show all four terms before collecting like terms.
(x + 4)(x + 7) =
(x + 2)(x - 9) =
(x - 5)(x - 3) =
(x - 8)(x + 8) =
Perfect Squares
Expand these perfect square expressions. Use the identity (a + b)^2 = a^2 + 2ab + b^2.
(x + 5)^2 =
(x - 4)^2 =
(x + 10)^2 =
(2x + 3)^2 =
Difference of Two Squares
Expand these using (a + b)(a - b) = a^2 - b^2.
(x + 6)(x - 6) =
(x + 9)(x - 9) =
(2x + 5)(2x - 5) =
(3x + 4)(3x - 4) =
Mixed Expansion Challenge
Expand and simplify each expression fully.
(x + 3)(x - 3) + x^2 =
2(x + 1)(x + 4) =
(x + 2)^2 - (x - 2)^2 =
Geometric Area Model
A rectangle has side lengths (x + 4) cm and (x + 3) cm.
Write an expression for the area as a binomial product.
Expand your expression to find the area as a trinomial.
If x = 5, what is the area of the rectangle in cm squared?
Binomial Expansion in Design
Try these real-world expansion activities at home.
- 1Draw a square with side length (x + 3) on grid paper. Shade the four regions that match (x + 3)^2 = x^2 + 6x + 9.
- 2Ask a family member to choose a two-digit number. Use difference of two squares to mentally calculate: e.g. 19 x 21 = (20-1)(20+1) = 399.
FOIL Method Practice
Use FOIL to expand each product. Label each step: F (First), O (Outer), I (Inner), L (Last).
(x + 6)(x + 2): F = ___, O = ___, I = ___, L = ___, Answer =
(x - 3)(x + 7): F = ___, O = ___, I = ___, L = ___, Answer =
(2x + 4)(x - 5): F = ___, O = ___, I = ___, L = ___, Answer =
Expanding (a + b)^2
Use the identity (a + b)^2 = a^2 + 2ab + b^2 to expand. State a and b for each.
(x + 7)^2: a = ___, b = ___, expansion =
(x + 12)^2: a = ___, b = ___, expansion =
(3x + 2)^2: a = ___, b = ___, expansion =
Expanding (a - b)^2
Use the identity (a - b)^2 = a^2 - 2ab + b^2 to expand.
(x - 7)^2 =
(x - 11)^2 =
(2x - 5)^2 =
(4x - 3)^2 =
Difference of Two Squares Pattern
Draw a line matching each product of the form (a + b)(a - b) to its expanded form a^2 - b^2.
Mixed Binomial Expansion with Negatives
Expand and simplify, taking care with negative signs.
(x - 4)(x - 9) =
(-x + 3)(x + 3) =
(x - 8)^2 =
(3 - x)(3 + x) =
Expanding with Coefficients
Expand fully. Take care when there is a coefficient on x.
(2x + 3)(x + 4) =
(3x - 2)(x - 5) =
(2x + 1)(2x - 1) =
(3x + 4)(3x + 4) =
Area Model Verification
For each binomial product, draw a rectangle with labelled sides, shade the four regions and write their areas, then expand algebraically and verify the answers match.
(x + 2)(x + 6): Draw rectangle, label 4 areas, expand algebraically.
(x + 5)(x - 1): Draw rectangle, label 4 areas, expand algebraically.
Why does the area model always produce exactly 4 partial products?
Spot the Error
Circle the correct expansion and cross out the wrong one.
(x + 4)^2 = ?
(x + 3)(x - 3) = ?
(2x + 1)^2 = ?
(x - 7)^2 = ?
Expand, Then Evaluate
First expand the expression, then substitute the given value of x to evaluate.
(x + 5)(x - 2) when x = 3: expand first, then substitute.
(x - 4)^2 when x = 6: expand first, then substitute.
(x + 10)(x - 10) when x = 10: expand first, then substitute.
Match to the Pattern Used
Match each expansion to the name of the identity pattern used.
Binomial Products in Geometry
A square swimming pool has side length (x + 6) metres. A path of width 2 m is added all around.
Write the side length of the outer square (pool + path).
Expand: (x + 10)^2 to find the area of the outer square.
Expand: (x + 6)^2 to find the area of the pool.
Write an expression for the area of the path only.
If x = 4, what is the area of the path?
Expanding Trinomial Expressions
These involve expanding first, then simplifying with additional terms.
(x + 4)(x - 2) + 5x =
(x - 3)^2 - 4x =
(x + 5)(x - 5) + 2x^2 =
2(x + 3)(x - 3) + (x + 1)^2 =
Applications: Perimeter and Area
A rectangular garden has length (2x + 5) m and width (x + 2) m.
Write an expression for the perimeter of the garden.
Write and expand an expression for the area of the garden.
If x = 3, find the area in square metres.
If the area is 54 m^2, write an equation and solve for x.
Prove the Identity
Prove each special product identity by expanding the left side using FOIL and collecting like terms.
Prove: (a + b)^2 = a^2 + 2ab + b^2. Start with (a+b)(a+b) and expand.
Prove: (a - b)^2 = a^2 - 2ab + b^2. Start with (a-b)(a-b) and expand.
Prove: (a + b)(a - b) = a^2 - b^2. Start with FOIL and show the middle terms cancel.
Which is Difference of Two Squares?
Circle the expression that is a difference of two squares.
Which fits (a+b)(a-b)?
Which is a perfect square?
Which product gives x^2 - 49?
Which product gives 9x^2 - 25?
Expanding Products with Fractions
Expand each product. Take care with fractional coefficients.
(x + 1/2)(x + 3/2) =
(x - 1/3)(x + 1/3) =
(2x + 1/2)^2 =
Sort by Expansion Type
Place each expression in the correct column.
Expanding Products with Two Variables
Expand each product involving two variables.
(x + y)(x + 2y) =
(2x - y)(x + 3y) =
(x + y)^2 =
(x - y)(x + y) =
Mental Maths Using Binomial Identities
Use these algebraic tricks for fast mental arithmetic.
- 1Calculate 31 x 29 using (30+1)(30-1) = 900 - 1 = 899. Verify with a calculator.
- 2Calculate 51^2 using (50+1)^2 = 2500 + 100 + 1 = 2601. Verify with a calculator.
- 3Calculate 47 x 53 using (50-3)(50+3) = 2500 - 9 = 2491. Verify with a calculator.
- 4Try 202^2 using (200+2)^2. Can you compute it mentally?
Expansion Challenge: Three Factors
Expand these products of three factors. Expand the first two brackets first, then multiply by the third.
(x + 1)(x + 2)(x + 3): Expand first two, then multiply by (x+3).
(x + 2)^2 (x - 1): Expand (x+2)^2 first, then multiply by (x-1).
x(x + 3)(x - 3): What shortcut can you use for (x+3)(x-3)?
Consecutive Integers Proof
Use binomial expansion to prove these results about consecutive integers.
Let two consecutive integers be n and n+1. Show that their product plus 1 is always a perfect square. Hint: expand n(n+1) + 1.
Show that the product of any two integers differing by 2 is always 1 less than the square of the number between them. Hint: let the integers be n-1 and n+1.
Link to Factorisation Preview
These expanded forms are quadratics. Recognise which special product pattern each came from.
x^2 - 25. What two binomials multiply to give this? What pattern is it?
x^2 + 16x + 64. What two identical binomials multiply to give this? What pattern is it?
x^2 - 10x + 25. What two identical binomials multiply to give this? What pattern is it?
4x^2 - 9. What pattern is this and what are the factors?
Expanding Products with Decimals
Expand these products that involve decimal coefficients.
(x + 0.5)(x + 0.5) =
(x + 0.1)(x - 0.1) =
(1.5x + 2)(x - 1) =
Error Analysis
Each expansion below contains exactly one error. Find the error and write the correct expansion.
Student writes: (x + 5)(x + 3) = x^2 + 15x + 15. Error:
Student writes: (x - 4)^2 = x^2 - 16. Error:
Student writes: (x + 7)(x - 7) = x^2 + 14x - 49. Error:
Student writes: (2x + 3)(x - 1) = 2x^2 - 5x - 3. Error:
Expanding Binomials: Open Investigation
Investigate the following pattern and write a general rule.
Expand: (x+1)(x-1), (x+2)(x-2), (x+3)(x-3), (x+4)(x-4). Write all four answers.
Describe the pattern in words: the product (x+n)(x-n) always equals ___
Why does the 'middle term' always disappear in these products?
Use the pattern to calculate 1004 x 996 mentally. Show your working.
Expanding Binomials — Exam Practice
These are exam-style questions. Set a timer for 20 minutes and attempt without assistance.
Expand and simplify: (x + 9)(x - 4)
Expand: (3x - 5)^2
Expand: (4x + 3)(4x - 3)
Expand and simplify: (x + 3)(x - 2) + (x - 1)^2
A rectangle has length (2x + 7) m and width (x - 1) m. Find an expression for the area, then find the area when x = 5.
Binomial Expansion Mastery Check
Complete each question to demonstrate your mastery of binomial expansion.
State the FOIL method in your own words.
State the three special product identities with an example of each.
Expand: (5x - 3)^2
Expand: (x + 2)(x^2 + 3x - 1)
Find two binomials whose product is x^2 - 100 and expand to verify.
What is the connection between binomial expansion and factorisation? Why does knowing one help with the other?
Worded Algebra Problems
Write and expand a binomial expression for each scenario.
A rectangle is x cm wide. Its length is 5 cm more than twice its width. Write and expand an expression for its area.
The side of a square is (x - 3) m. Write and expand an expression for the square's area.
A border of width 2 m surrounds a square photo of side length (x + 2) m. Express the total framed area, expanded.
Expand (x + 2)^3
Expand using repeated FOIL, then answer the questions.
Expand (x + 2)^2 first:
Now multiply your result by (x + 2):
List the coefficients: 1, ___, ___, ___. Do you recognise Pascal's Triangle row 3?
Middle Term Match
Match each product to its middle term (x-term) after FOIL expansion.
Reverse Engineering — Find the Binomials
Work backwards from the expanded form to find the binomial factors.
x^2 + 10x + 21 = ( ___ )( ___ ). Verify by expanding.
x^2 - 4x - 12 = ( ___ )( ___ ). Verify by expanding.
x^2 - 18x + 81 = ( ___ )^2. Verify by expanding.
Is it possible for a binomial product to give x^2 + 1? Explain.
Connecting Expansion to Graphing
Each expansion below is a quadratic. Describe what you can read from the expanded form.
Expand (x + 3)(x - 1). What is the y-intercept of y = (x+3)(x-1)? What are the x-intercepts?
Expand (x - 4)^2. Does the graph of y = (x-4)^2 touch or cross the x-axis? At what x?
Expand (x + 5)(x - 5). What is special about this parabola?
Binomials in Daily Life
Explore where binomial products appear in everyday contexts.
- 1Measure a room. Write its dimensions as (x + a) and (y + b) where x, y are base lengths. Expand to find an area formula.
- 2A garden path 1 m wide surrounds a square garden of side x m. Write the total area as a binomial product and expand. Verify numerically for x = 5.
- 3Use the difference of two squares trick to mentally calculate: 997 x 1003, 48 x 52, 63 x 57. Write your working.
Error Analysis
Each expansion below contains exactly one error. Find it and write the correction.
Student writes: (x + 5)(x + 3) = x^2 + 15x + 15. Error:
Student writes: (x - 4)^2 = x^2 - 16. Error:
Student writes: (x + 7)(x - 7) = x^2 + 14x - 49. Error:
Student writes: (2x + 3)(x - 1) = 2x^2 - 5x - 3. Error:
Investigation: Products of Consecutive Integers
Investigate the relationship between consecutive integer products and perfect squares.
Let n be any integer. Expand n(n+1) + 1. What do you notice about the result?
Let consecutive integers be n-1 and n+1. Expand (n-1)(n+1). How does it relate to n^2?
Use this result to explain: the product of any two integers differing by 2 is always 1 less than the square of the integer between them.
Binomial Products — Final Cumulative Review
Exam practice. Work independently and show all steps.
Expand: (x + 8)(x - 3)
Expand: (4x - 1)^2
Expand: (6x + 5)(6x - 5)
Expand and simplify: (x + 5)^2 - (x - 5)^2
A rectangle has length (2x + 7) m and width (x - 1) m. Expand and find the area when x = 5.
Explain in one sentence why (a + b)^2 ≠ a^2 + b^2.
Correct Expansion
Circle the correctly expanded expression.
(x + 6)(x - 4)
(x - 5)^2
(2x + 3)(2x - 3)
(x + 7)^2
Expand Products — Increasing Difficulty
Expand each expression. These increase in difficulty.
(x + 3)(x - 8)
(2x - 5)(x + 4)
(3x + 2)^2 - (x - 1)^2
(x + a)(x - a) - (x + a)^2. Simplify fully in terms of a and x.
Binomials in Financial Mathematics
A company's revenue is modelled by (x + 100)(x - 20) dollars, where x is the number of units sold.
Expand the revenue expression.
What is the revenue when x = 50?
At what values of x is revenue equal to zero? What do these values represent?
Write a similar revenue model for a different product and expand it.
Create and Expand a Binomial Product Story
Create a real-world story problem involving a binomial product, write the algebraic expression, and expand it.
Write a scenario that produces a binomial product (e.g. area, perimeter, cost).
Write the algebraic binomial product for your scenario.
Expand the expression and interpret each term in the context of your story.
Substitute a value for x and calculate the numerical answer for your story.
Binomial Products — Comprehensive Final
These ten questions cover the full range of binomial expansion skills. Show all working.
1. (x + 6)(x + 4) =
2. (x - 9)(x + 2) =
3. (x - 5)^2 =
4. (x + 11)(x - 11) =
5. (2x + 3)(x - 4) =
6. (3x - 2)^2 =
7. (5x + 1)(5x - 1) =
8. (x + 4)(x - 4) + (x + 4)^2 =
9. A square has side (2x - 3) cm. Find an expression for its area.
10. Prove that the product of n and (n+2), added to 1, is always a perfect square. (Let n be any integer.)
Expanding with Three Terms in One Factor
Expand each product where one factor has three terms.
(x + 2)(x^2 + 3x + 1) =
(x - 1)(x^2 - x + 1) =
What pattern do you notice in (x - 1)(x^2 + x + 1)? Compute it and describe the result.
Which Product Has a Positive Constant?
Circle the binomial product whose constant term is positive after expansion.
Which has a positive constant?
Which has a positive constant?
Expanding in Number Theory
Use binomial expansion to explore patterns in number theory.
Show algebraically that the sum of two consecutive squares minus the square of their average is always 1/2. Hint: use (n-1/2)^2.
Show that (n+1)^2 - n^2 = 2n + 1. What does this prove about consecutive squares?
Use this result to calculate 51^2 - 50^2 mentally.
Binomial Expansion — Self-Reflection
Reflect on your mastery of binomial expansion before moving to factorisation.
Which type of expansion do you find easiest? Why?
Which type do you find hardest? What specific step causes the most difficulty?
Expand three examples of the type you find hardest, right now, from scratch.
How confident are you (1–10) that you can expand any binomial product in an exam? What would raise your confidence?
Match Expansion to Its Type
Match each expansion to the correct identity name.
Spot, State, Expand
For each product, name the pattern, state the identity you will use, then expand.
(x + 15)(x - 15): Pattern = ___, Expansion =
(x - 9)^2: Pattern = ___, Expansion =
(2x + 5)^2: Pattern = ___, Expansion =
(x + 4)(x - 7): Pattern = ___, Expansion =
Binomial Products — Ten-Question Sprint
Expand each of these ten products as fast as you can. Aim for under 10 minutes.
1. (x + 2)(x + 7) = 2. (x - 3)(x + 6) = 3. (x - 5)^2 =
4. (x + 8)(x - 8) = 5. (2x + 3)(x + 1) = 6. (4x - 5)^2 =
7. (3x + 7)(3x - 7) = 8. (x + 4)(x - 9) = 9. (x - 6)^2 =
10. (5x + 2)(2x - 3) =
Completing the Square Preview
Rewrite each expression in the form (x + h)^2 + k by completing the square.
x^2 + 6x + 5 = (x + ___)^2 - ___ + 5 = (x + ___)^2 + ___
x^2 + 8x - 3 = (x + ___)^2 - ___ - 3 = (x + ___)^2 - ___
x^2 - 4x + 1 = (x - ___)^2 - ___ + 1 = (x - ___)^2 - ___
Largest Expansion Value at x=3
Evaluate each expression at x = 3 and circle the one with the largest value.
Largest at x = 3?
Largest at x = 3?
Expansion Quick-Fire Round
Expand these four expressions as quickly as possible. No FOIL label needed — just write the final expanded form.
(x + 12)(x - 12) =
(x - 10)^2 =
(3x + 5)(x - 2) =
(2x - 9)^2 =