3. Expanding Binomials

Expanding Binomials

Introduction

Algebra often involves expressions containing two or more terms. Learning how to multiply these expressions is an essential skill for working with polynomials, solving equations, and understanding quadratic functions.

A binomial is an algebraic expression containing two terms.

Examples include:

• x + 3

• x − 5

• 2x + 7

• 3a − 4b

When two binomials are multiplied, we use a process called expansion.

For example:

(x + 2)(x + 3)

Expanding this expression gives:

x² + 5x + 6

The expanded expression is equivalent to the original product, but it is written as a sum of terms rather than a multiplication of factors.

Expanding binomials is important because it allows us to:

• Simplify algebraic expressions.

• Develop quadratic equations.

• Understand polynomial multiplication.

• Calculate areas of geometric shapes.

• Verify factorisation.

• Solve practical problems involving changing quantities.

What Is a Binomial?

A binomial is a polynomial containing exactly two terms.

Examples:

x + 4

Terms: x and 4

3x − 7

Terms: 3x and −7

2a + 5b

Terms: 2a and 5b

x² − 9

Terms: x² and −9

A binomial may contain variables, coefficients, constants, and powers.

Important Observation

A binomial contains two terms, but those terms do not have to be simple numbers or variables.

Understanding Multiplication of Binomials

Consider:

(x + 2)(x + 3)

This means:

(x + 2) multiplied by (x + 3).

Each term in the first binomial must multiply each term in the second binomial.

There are four individual products:

• x × x = x²

• x × 3 = 3x

• 2 × x = 2x

• 2 × 3 = 6

Add the results:

x² + 3x + 2x + 6

Combine like terms:

x² + 5x + 6

Important Observation

Multiplying two binomials normally produces four intermediate products, although some terms can be combined.

The Distributive Property

The distributive property is the foundation of expanding algebraic expressions.

For numbers:

a(b + c) = ab + ac

For example:

3(4 + 5) = 3 × 4 + 3 × 5

= 12 + 15

= 27

The same rule applies to algebra.

Example

Expand:

2x(x + 4)

Distribute 2x:

2x × x + 2x × 4

= 2x² + 8x

Important Observation

Every term inside the brackets must be multiplied by the term outside.

The Double Distributive Property

When multiplying two binomials, we apply the distributive property twice.

Worked Example

Expand:

(x + 2)(x + 5)

Step 1: Distribute the First Term

x(x + 5) = x² + 5x

Step 2: Distribute the Second Term

2(x + 5) = 2x + 10

Step 3: Combine

x² + 5x + 2x + 10

Step 4: Collect Like Terms

x² + 7x + 10

This method works for all products of binomials.

The Area Model for Expanding Binomials

Multiplication of binomials can also be understood geometrically.

Imagine a rectangle with side lengths:

x + 3 and x + 2

Its area is:

A = (x + 3)(x + 2)

We can divide the rectangle into four smaller regions.

The total area is:

A = x² + 3x + 2x + 6

Combine like terms:

A = x² + 5x + 6

Therefore:

(x + 3)(x + 2) = x² + 5x + 6

Important Observation

The area model helps explain why each term in one binomial must multiply every term in the other binomial.

The FOIL Method

FOIL is a useful memory aid for multiplying two binomials.

FOIL stands for:

• F — First: Multiply the first terms.

• O — Outer: Multiply the outer terms.

• I — Inner: Multiply the inner terms.

• L — Last: Multiply the last terms.

Worked Example

Expand:

(x + 4)(x + 7)

First:

x × x = x²

Outer:

x × 7 = 7x

Inner:

4 × x = 4x

Last:

4 × 7 = 28

Combine:

x² + 7x + 4x + 28

Final Answer: x² + 11x + 28

Important Observation

FOIL is a convenient method for multiplying two binomials. The distributive property is more general and also works for expressions containing more than two terms.

Organising Terms Systematically

A systematic approach reduces errors.

Worked Example

Expand:

(2x + 3)(x + 5)

Add:

2x² + 10x + 3x + 15

Combine like terms:

2x² + 13x + 15

Important Observation

Writing all four products before simplifying helps prevent missing terms.

Expanding Binomials with Positive Terms

Worked Example

Expand:

(x + 6)(x + 2)

First:

x × x = x²

Outer:

x × 2 = 2x

Inner:

6 × x = 6x

Last:

6 × 2 = 12

Combine:

x² + 2x + 6x + 12

Final Answer: x² + 8x + 12

Another Example

Expand:

(3x + 4)(2x + 5)

First:

3x × 2x = 6x²

Outer:

3x × 5 = 15x

Inner:

4 × 2x = 8x

Last:

4 × 5 = 20

Combine:

6x² + 23x + 20

Expanding Binomials with Negative Terms

Negative signs require careful attention.

Worked Example

Expand:

(x − 3)(x + 5)

First:

x × x = x²

Outer:

x × 5 = 5x

Inner:

−3 × x = −3x

Last:

−3 × 5 = −15

Combine:

x² + 5x − 3x − 15

Final Answer: x² + 2x − 15

Important Observation

Treat subtraction as addition of a negative term.

This helps avoid sign errors.

Expanding Two Binomials with Negative Terms

Worked Example

Expand:

(x − 4)(x − 6)

First:

x × x = x²

Outer:

x × (−6) = −6x

Inner:

−4 × x = −4x

Last:

−4 × (−6) = +24

Combine:

x² − 6x − 4x + 24

Final Answer: x² − 10x + 24

Important Observation

The product of two negative numbers is positive.

Sign Rules in Binomial Expansion

Remember the multiplication rules:

Example

Expand:

(2x − 5)(3x − 4)

Products:

2x × 3x = 6x²

2x × (−4) = −8x

−5 × 3x = −15x

−5 × (−4) = +20

Combine:

6x² − 8x − 15x + 20

Final Answer: 6x² − 23x + 20

Combining Like Terms

After expansion, some terms can be combined.

Like terms have the same variables raised to the same powers.

Examples:

• 3x and 7x are like terms.

• 5x² and −2x² are like terms.

• 4ab and 9ab are like terms.

However:

• x and x² are not like terms.

• x and y are not like terms.

• xy and x² are not like terms.

Worked Example

Simplify:

3x² + 5x + 2x² − 7x + 4

Group like terms:

(3x² + 2x²) + (5x − 7x) + 4

Simplify:

5x² − 2x + 4

Important Observation

Only terms with identical variable parts can be combined by addition or subtraction.

Expanding Binomials with Different Variables

Binomials may contain more than one variable.

Worked Example

Expand:

(a + b)(a + 3b)

First:

a × a = a²

Outer:

a × 3b = 3ab

Inner:

b × a = ab

Last:

b × 3b = 3b²

Combine:

a² + 3ab + ab + 3b²

Final Answer: a² + 4ab + 3b²

Another Example

Expand:

(2x − y)(3x + 4y)

Products:

2x × 3x = 6x²

2x × 4y = 8xy

−y × 3x = −3xy

−y × 4y = −4y²

Combine:

6x² + 5xy − 4y²

Expanding Binomials Containing Powers

The terms inside a binomial may already contain powers.

Worked Example

Expand:

(x² + 3)(x + 4)

First:

x² × x = x³

Outer:

x² × 4 = 4x²

Inner:

3 × x = 3x

Last:

3 × 4 = 12

Therefore:

x³ + 4x² + 3x + 12

Worked Example

Expand:

(2x² − 3x)(x + 5)

Products:

2x² × x = 2x³

2x² × 5 = 10x²

−3x × x = −3x²

−3x × 5 = −15x

Combine:

2x³ + 7x² − 15x

Important Observation

When multiplying powers with the same base, add their exponents.

For example:

x² × x³ = x⁵

The General Binomial Product

The distributive property can be expressed using a general formula.

(a + b)(c + d) = ac + ad + bc + bd

This formula works for numbers, variables, and algebraic expressions.

Example

Expand:

(2x + 5)(3x + 7)

Using the general formula:

= (2x)(3x) + (2x)(7) + (5)(3x) + (5)(7)

= 6x² + 14x + 15x + 35

= 6x² + 29x + 35

Important Observation

FOIL is simply a specific way of applying this general distributive pattern.

Using an Algebraic Grid

An algebraic grid provides another systematic method for multiplication.

Worked Example

Expand:

(2x + 3)(x + 4)

Add the four products:

2x² + 8x + 3x + 12

Combine:

2x² + 11x + 12

Advantages of the Grid Method

• Organises every multiplication.

• Reduces the chance of missing terms.

• Makes signs easier to track.

• Extends naturally to larger polynomials.

Important Observation

The grid method and FOIL produce identical results because both use the distributive property.

Special Case: Squaring a Binomial

Sometimes the two binomials are identical.

Example

(x + 4)(x + 4)

This can be written:

(x + 4)²

Expand:

x² + 4x + 4x + 16

= x² + 8x + 16

This is a perfect square trinomial.

General Identity

(a + b)² = a² + 2ab + b²

Another Example

(x − 5)² = x² − 10x + 25

Important Observation

The square of a binomial includes a middle term. It is not simply the sum of the squares of its terms.

Special Case: Conjugate Binomials

Conjugate binomials contain the same terms but opposite signs.

For example:

(x + 4)(x − 4)

Expand:

x² − 4x + 4x − 16

The middle terms cancel.

= x² − 16

This is called the difference of two squares.

General Identity

(a + b)(a − b) = a² − b²

Another Example

(3x + 5)(3x − 5)

= 9x² − 25

Important Observation

Recognising conjugate binomials allows us to expand without writing all four intermediate products.

Comparing Common Expansion Patterns

These patterns are useful shortcuts, but they all come from the distributive property.

Verifying Expansions by Factorisation

Expansion and factorisation are inverse operations.

Worked Example

Expand:

(x + 3)(x + 5)

Result:

x² + 8x + 15

To verify, factor the expanded expression.

We need two numbers that:

• Multiply to 15.

• Add to 8.

The numbers are 3 and 5.

Therefore:

x² + 8x + 15 = (x + 3)(x + 5)

The expansion is correct.

Important Observation

Factoring the result back into the original binomials is one way to verify an expansion.

Verifying Expansions by Substitution

A numerical check can help detect mistakes.

Worked Example

Verify:

(x + 2)(x + 6) = x² + 8x + 12

Choose x = 3.

Original Expression

(3 + 2)(3 + 6)

= 5 × 9

= 45

Expanded Expression

3² + 8(3) + 12

= 9 + 24 + 12

= 45

Both expressions produce the same result.

Important Observation

Substitution is useful for checking an expansion, but a single numerical test does not prove an identity.

Algebraic expansion provides a complete verification.

Verifying with a Second Method

A particularly reliable strategy is to expand using one method and check using another.

Example

Expand:

(3x − 2)(2x + 5)

Using FOIL:

6x² + 15x − 4x − 10

= 6x² + 11x − 10

Check with an algebraic grid:

The same four terms appear.

Therefore, the expansion is correct.

Applications in Geometry

Binomial multiplication is useful for calculating the areas of rectangles.

Worked Example

A rectangular garden has:

Length = x + 7 metres

Width = x + 3 metres

Find an expression for its area.

Area = Length × Width

A = (x + 7)(x + 3)

Expand:

A = x² + 3x + 7x + 21

A = x² + 10x + 21

Interpretation

The area depends on the value of x.

If x = 5:

A = 5² + 10(5) + 21

= 25 + 50 + 21

A = 96 m²

Applications in Number Calculations

Binomial multiplication can simplify mental arithmetic.

Worked Example: 23 × 14

Rewrite:

23 = 20 + 3

14 = 10 + 4

Therefore:

23 × 14 = (20 + 3)(10 + 4)

Expand:

= 200 + 80 + 30 + 12

= 322

Worked Example: 48 × 52

Rewrite:

48 = 50 − 2

52 = 50 + 2

Apply the difference-of-squares identity:

(50 − 2)(50 + 2)

= 50² − 2²

= 2500 − 4

= 2496

Important Observation

Recognising useful binomial patterns can make calculations faster and more efficient.

Applications in Science

Binomial expansion appears when scientific equations contain sums or differences.

Example: Kinetic Energy

The kinetic energy of an object is:

Eₖ = ½mv²

Suppose velocity is:

v = u + at

Substitute:

Eₖ = ½m(u + at)²

Expand:

(u + at)² = u² + 2uat + a²t²

Therefore:

Eₖ = ½mu² + muat + ½ma²t²

This expression shows how kinetic energy depends on initial velocity, acceleration, and time when acceleration is constant.

Important Observation

Algebraic expansion is essential for rearranging and analysing many scientific equations.

Applications in Quadratic Functions

Multiplying two linear factors often produces a quadratic expression.

Example

Consider:

y = (x + 2)(x + 4)

Expand:

y = x² + 6x + 8

This is a quadratic function.

Its graph is a parabola.

The original factored form shows that the function equals zero when:

x = −2 or x = −4

The expanded form shows the coefficients of the quadratic.

Important Observation

Factored and expanded forms provide different useful information about the same quadratic function.

Common Mistakes

Forgetting One of the Four Products

Incorrect:

(x + 2)(x + 3) = x² + 6

Correct:

(x + 2)(x + 3) = x² + 5x + 6

Multiplying Only the First and Last Terms

Every term in the first binomial must multiply every term in the second binomial.

Incorrect Sign Rules

Incorrect:

(x − 4)(x − 2) = x² − 6x − 8

Correct:

(x − 4)(x − 2) = x² − 6x + 8

Combining Unlike Terms

Incorrect:

x² + 3x = 4x²

Correct:

x² + 3x cannot be combined into one term.

Forgetting Exponent Rules

Incorrect:

x × x = 2x

Correct:

x × x = x²

Forgetting the Middle Term When Squaring

Incorrect:

(x + 5)² = x² + 25

Correct:

(x + 5)² = x² + 10x + 25

Not Simplifying Completely

An expansion should normally be simplified by combining all like terms.

Failing to Verify

Use substitution, factorisation, or a second multiplication method to check your result.

Key Terms

• Binomial: Algebraic expression containing two terms.

• Polynomial: Algebraic expression formed from terms containing variables with nonnegative integer powers.

• Expansion: Rewriting a product as a sum of terms.

• Distributive Property: Rule that distributes multiplication across addition or subtraction.

• Double Distribution: Applying the distributive property to multiply two binomials.

• FOIL: Method involving First, Outer, Inner, and Last products.

• Term: Part of an algebraic expression separated by addition or subtraction.

• Coefficient: Numerical factor multiplying a variable.

• Constant: Term without a variable.

• Like Terms: Terms with identical variable parts and exponents.

• Simplification: Rewriting an expression in an equivalent, more convenient form.

• Algebraic Grid: Table used to organise polynomial multiplication.

• Area Model: Geometric representation of multiplication using rectangular regions.

• Perfect Square Trinomial: Trinomial formed by squaring a binomial.

• Conjugate Binomials: Binomials with the same terms and opposite signs.

• Difference of Squares: Identity (a + b)(a − b) = a² − b².

• Factorisation: Rewriting an expression as a product.

• Verification: Checking that a mathematical result is correct.

• Quadratic Expression: Polynomial expression with highest variable power 2.

• Quadratic Function: Function represented by a second-degree polynomial.

Key Takeaways

• A binomial contains two terms.

• Expanding means multiplying algebraic expressions and rewriting the result as a sum.

• Every term in one binomial must multiply every term in the other.

• FOIL is a useful method for multiplying two binomials.

• The distributive property is the foundation of binomial expansion.

• Algebraic grids help organise multiplication systematically.

• Negative signs must be handled carefully.

• Like terms should be combined after expansion.

• When multiplying powers with the same base, add the exponents.

• Squaring a binomial produces a perfect square trinomial.

• Multiplying conjugate binomials produces a difference of squares.

• Expansion and factorisation are inverse processes.

• Substitution can help check an expansion.

• Binomial expansion is useful in geometry, arithmetic, science, and quadratic functions.

Check Your Understanding

Basic Expansion

1. What is a binomial?

2. What does FOIL stand for?

3. Expand (x + 2)(x + 4).

4. Expand (x + 3)(x + 7).

5. Expand (x + 5)(x + 6).

Negative Terms

6. Expand (x − 2)(x + 5).

7. Expand (x − 4)(x − 3).

8. Expand (x + 6)(x − 2).

9. Expand (2x − 3)(x + 4).

10. Expand (3x − 2)(2x − 5).

Coefficients and Variables

11. Expand (2x + 3)(3x + 4).

12. Expand (4x + 1)(2x + 5).

13. Expand (a + b)(a + 2b).

14. Expand (2x − y)(x + 3y).

15. Expand (x² + 2)(x + 3).

Special Patterns and Verification

16. Expand (x + 5)².

17. Expand (x − 7)².

18. Expand (x + 4)(x − 4).

19. Explain why (x + 3)² is not equal to x² + 9.

20. Describe two methods for verifying an expansion.

Answer Key

Extended Challenge

1. Expand (4x + 7)(3x + 2).

2. Expand (5a − 3b)(2a + 4b).

3. Expand (x² + 3x)(2x − 5).

4. Expand (2x² − 3)(x² + 4).

5. Expand (3x − 4)².

6. Expand (5a + 2b)².

7. Expand (4x + 7)(4x − 7).

8. Verify by expansion that (x + 2)(x + 6) = x² + 8x + 12.

9. A rectangle has dimensions 2x + 3 and x + 5. Find an expression for its area.

10. Explain why multiplying two first-degree binomials usually produces a quadratic expression.

Extended Challenge Answers

Final Investigation: Designing a Rectangular Garden

A school plans to create a rectangular garden.

The length is:

x + 6 metres

The width is:

x + 4 metres

Part A: Building the Expression

1. Write an expression for the garden's area.

2. Expand the expression using FOIL.

3. Simplify the result.

4. Expand the expression using an algebraic grid.

5. Compare the two results.

Part B: Creating a Table

6. Calculate the area when x = 1, 2, 3, 4, and 5.

7. Organise the results in a table.

8. Determine how the area changes as x increases.

9. Explain whether the relationship is linear or nonlinear.

10. Explain why the expanded expression contains x².

Part C: Applying the Model

11. Calculate the garden's area when x = 6.

12. Calculate the garden's area when x = 10.

13. Find the difference between these two areas.

14. Explain how increasing x changes both dimensions.

15. Explain why the area increases faster as x becomes larger.

Part D: Verification

16. Choose one value of x.

17. Calculate the area directly using length × width.

18. Calculate the area using the expanded expression.

19. Compare the results.

20. Explain why the two methods must agree.

Investigation Answer Check

Original Expression

A = (x + 6)(x + 4)

Expansion

A = x² + 4x + 6x + 24

A = x² + 10x + 24

Completed Table

The area increases by:

13, 15, 17, 19 m²

The differences are not constant.

Therefore, the relationship is nonlinear.

When x = 6

A = 6² + 10(6) + 24

= 36 + 60 + 24

A = 120 m²

When x = 10

A = 10² + 10(10) + 24

= 100 + 100 + 24

A = 224 m²

Difference in Area

224 − 120 = 104 m²

Verification

When x = 6:

Length = 12 m

Width = 10 m

Area = 12 × 10

Area = 120 m²

This agrees with the expanded expression.

Final Reflection

Expanding binomials is an essential algebraic skill that connects multiplication, simplification, factorisation, geometry, and quadratic functions.

By applying the distributive property systematically, we can multiply algebraic expressions accurately and simplify the results.

FOIL, algebraic grids, and area models are different ways of organising the same mathematical process.

The most important principle is that every term in one binomial must multiply every term in the other binomial, and all like terms must then be combined correctly.