Multiplying Polynomials

Introduction

Multiplying polynomials is an important algebraic skill used to simplify expressions, solve equations, calculate areas, and model real-world situations.

Unlike adding and subtracting polynomials, multiplication often creates new terms with higher powers of variables.

For example:

3x × 4x = 12x²

And:

(x + 2)(x + 3) = x² + 5x + 6

To multiply polynomials correctly, we use the distributive property, exponent rules, and the process of combining like terms.

In this topic, we will explore multiplication involving monomials, binomials, trinomials, and more complicated polynomial expressions.

Review: What Is a Polynomial?

A polynomial is an algebraic expression consisting of one or more terms containing numbers, variables, and nonnegative integer exponents.

Examples include:

• 5x

• 3x + 7

• 2x² − 4x + 5

• x³ + 2x² − x + 9

Polynomials are classified according to their number of terms.

When multiplying polynomials, each term in one expression must be multiplied by every term in the other expression.

Review: Multiplication of Powers

Before multiplying polynomials, we need to understand how exponents behave during multiplication.

Product of Powers Rule

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

xᵃ × xᵇ = xᵃ⁺ᵇ

For example:

x² × x³ = x⁵

x⁴ × x = x⁵

x³ × x³ = x⁶

Multiplying Coefficients

When terms contain numerical coefficients, multiply the coefficients separately.

For example:

3x² × 4x³

Multiply the coefficients:

3 × 4 = 12

Add the exponents:

x² × x³ = x⁵

Therefore:

3x² × 4x³ = 12x⁵

Another Example

−5a³ × 2a²

Multiply:

−5 × 2 = −10

Add the exponents:

a³ × a² = a⁵

Therefore:

−5a³ × 2a² = −10a⁵

Multiplying Monomials

A monomial is a polynomial containing only one term.

To multiply monomials:

1. Multiply the numerical coefficients.

2. Multiply variables with the same base by adding exponents.

3. Include any variables that appear in only one factor.

4. Simplify the result.

Worked Example

Multiply:

4x² × 3x⁴

Coefficients:

4 × 3 = 12

Variables:

x² × x⁴ = x⁶

Final answer: 12x⁶

Worked Example with Multiple Variables

Multiply:

3a²b × 4ab³

Coefficients:

3 × 4 = 12

Variables:

a² × a = a³

b × b³ = b⁴

Therefore:

3a²b × 4ab³ = 12a³b⁴

Worked Example with Negative Coefficients

Multiply:

−2x³y × −5xy²

Coefficients:

−2 × −5 = 10

Variables:

x³ × x = x⁴

y × y² = y³

Final answer: 10x⁴y³

Important Sign Rules

The Distributive Property

The distributive property allows multiplication to be applied to every term inside parentheses.

The general rule is:

a(b + c) = ab + ac

Similarly:

a(b − c) = ab − ac

This property is the foundation of polynomial multiplication.

Worked Example

Expand:

3(x + 4)

Multiply 3 by each term:

3 × x = 3x

3 × 4 = 12

Therefore:

3(x + 4) = 3x + 12

Worked Example with Subtraction

Expand:

5(2x − 3)

Multiply 5 by each term:

5 × 2x = 10x

5 × (−3) = −15

Therefore:

5(2x − 3) = 10x − 15

Multiplying a Monomial by a Polynomial

When multiplying a monomial by a polynomial, distribute the monomial to every term.

Worked Example

Expand:

4x(2x + 5)

Distribute:

4x × 2x = 8x²

4x × 5 = 20x

Therefore:

4x(2x + 5) = 8x² + 20x

Worked Example with a Trinomial

Expand:

3x(2x² − 4x + 5)

Multiply each term:

3x × 2x² = 6x³

3x × (−4x) = −12x²

3x × 5 = 15x

Therefore:

3x(2x² − 4x + 5) = 6x³ − 12x² + 15x

Worked Example with a Negative Monomial

Expand:

−2x(3x² − 5x + 4)

Distribute:

−2x × 3x² = −6x³

−2x × (−5x) = 10x²

−2x × 4 = −8x

Final answer: −6x³ + 10x² − 8x

Multiplying Two Binomials

A binomial contains two terms.

When multiplying two binomials, each term in the first binomial must multiply both terms in the second binomial.

Consider:

(x + 2)(x + 3)

Distribute x:

x(x + 3) = x² + 3x

Distribute 2:

2(x + 3) = 2x + 6

Combine:

x² + 3x + 2x + 6

Combine like terms:

x² + 5x + 6

General Pattern

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

There are four individual products before any like terms are combined.

Understanding Binomial Multiplication with an Area Model

A rectangle can be divided into four smaller regions.

If the rectangle has side lengths:

x + 2

and:

x + 3

its area is:

(x + 2)(x + 3)

The four smaller areas are:

• x × x = x²

• x × 3 = 3x

• 2 × x = 2x

• 2 × 3 = 6

Adding:

x² + 3x + 2x + 6

= x² + 5x + 6

The area model helps explain why every term must be multiplied.

The FOIL Method

FOIL is a memory aid for multiplying two binomials.

FOIL stands for:

• First: Multiply the first terms.

• Outer: Multiply the outer terms.

• Inner: Multiply the inner terms.

• 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

= x² + 11x + 28

Important Limitation

FOIL is useful only for multiplying two binomials.

The distributive property is more general and works for all polynomial products.

Multiplying Binomials with Negative Terms

Worked Example

Expand:

(x − 3)(x + 5)

Multiply:

x × x = x²

x × 5 = 5x

−3 × x = −3x

−3 × 5 = −15

Combine:

x² + 5x − 3x − 15

= x² + 2x − 15

Worked Example with Two Negative Terms

Expand:

(x − 4)(x − 6)

Multiply:

x² − 6x − 4x + 24

Combine:

x² − 10x + 24

Notice that:

−4 × −6 = +24

Multiplying Binomials with Coefficients

Not every binomial begins with x.

Worked Example

Expand:

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

Multiply each pair:

2x × 4x = 8x²

2x × 5 = 10x

3 × 4x = 12x

3 × 5 = 15

Combine:

8x² + 10x + 12x + 15

= 8x² + 22x + 15

Another Example

Expand:

(3x − 2)(2x − 7)

Multiply:

3x × 2x = 6x²

3x × (−7) = −21x

−2 × 2x = −4x

−2 × (−7) = 14

Combine:

6x² − 25x + 14

Multiplying a Binomial by a Trinomial

A binomial contains two terms, while a trinomial contains three.

Each term in the binomial must multiply all three terms in the trinomial.

Worked Example

Expand:

(x + 2)(x² + 3x + 4)

Distribute x:

x(x² + 3x + 4)

= x³ + 3x² + 4x

Distribute 2:

2(x² + 3x + 4)

= 2x² + 6x + 8

Combine:

x³ + 3x² + 4x + 2x² + 6x + 8

Group like terms:

x³ + (3x² + 2x²) + (4x + 6x) + 8

= x³ + 5x² + 10x + 8

Another Example

Expand:

(2x − 3)(x² − 4x + 5)

Distribute 2x:

2x³ − 8x² + 10x

Distribute −3:

−3x² + 12x − 15

Combine:

2x³ − 8x² + 10x − 3x² + 12x − 15

= 2x³ − 11x² + 22x − 15

Multiplying Two Trinomials

Multiplying two trinomials involves nine individual products before like terms are combined.

Worked Example

Expand:

(x² + 2x + 3)(x² + x + 4)

Distribute x²:

x⁴ + x³ + 4x²

Distribute 2x:

2x³ + 2x² + 8x

Distribute 3:

3x² + 3x + 12

Combine:

x⁴ + x³ + 4x² + 2x³ + 2x² + 8x + 3x² + 3x + 12

Group like terms:

x⁴ + (x³ + 2x³) + (4x² + 2x² + 3x²) + (8x + 3x) + 12

= x⁴ + 3x³ + 9x² + 11x + 12

Important Strategy

When multiplying larger polynomials, work systematically.

Multiply every term in the first polynomial by every term in the second polynomial.

Then combine like terms.

Polynomial Multiplication Using a Grid

A multiplication grid helps organise longer calculations.

Consider:

(2x + 3)(x² + 4x + 5)

Combine all entries:

2x³ + 8x² + 10x + 3x² + 12x + 15

= 2x³ + 11x² + 22x + 15

Advantages of the Grid Method

• Helps ensure every term is multiplied.

• Reduces the chance of missing products.

• Makes negative signs easier to track.

• Works for binomials, trinomials, and larger polynomials.

Common Polynomial Products

Some polynomial products occur so frequently that recognising their patterns can save time.

These are called special products or special product identities.

The three most important are:

Square of a sum:

(a + b)² = a² + 2ab + b²

Square of a difference:

(a − b)² = a² − 2ab + b²

Product of a sum and a difference:

(a + b)(a − b) = a² − b²

These identities come directly from the distributive property.

Squaring a Binomial: Square of a Sum

Consider:

(x + 3)²

The exponent 2 means the entire binomial is multiplied by itself.

(x + 3)² = (x + 3)(x + 3)

Distribute:

= x² + 3x + 3x + 9

Combine:

(x + 3)² = x² + 6x + 9

General Rule

(a + b)² = a² + 2ab + b²

The result contains:

• The square of the first term.

• Twice the product of the two terms.

• The square of the second term.

Worked Example

Expand:

(2x + 5)²

Let:

a = 2x

b = 5

Apply the identity:

(2x)² + 2(2x)(5) + 5²

= 4x² + 20x + 25

Final answer: 4x² + 20x + 25

Important Reminder

A common error is:

(x + 3)² = x² + 9

This is incorrect because it omits the middle term, 6x.

Squaring a Binomial: Square of a Difference

Consider:

(x − 4)²

Rewrite:

(x − 4)(x − 4)

Distribute:

x² − 4x − 4x + 16

Combine:

(x − 4)² = x² − 8x + 16

General Rule

(a − b)² = a² − 2ab + b²

Worked Example

Expand:

(3x − 2)²

Apply the identity:

(3x)² − 2(3x)(2) + 2²

= 9x² − 12x + 4

Final answer: 9x² − 12x + 4

Notice that the final term is positive because squaring a negative quantity produces a positive result.

Difference of Squares

The product of a sum and a difference produces a particularly useful pattern.

(a + b)(a − b) = a² − b²

The middle terms cancel.

Worked Example

Expand:

(x + 5)(x − 5)

Distribute:

x² − 5x + 5x − 25

Combine:

x² − 25

Another Example

Expand:

(3x + 4)(3x − 4)

Apply the identity:

(3x)² − 4²

= 9x² − 16

Final answer: 9x² − 16

Why Is This Useful?

Recognising the difference-of-squares pattern allows us to expand some expressions without writing every individual product.

Recognising Special Products

Compare these expressions.

The expressions look similar, but their products are different.

Always check whether the binomials are identical or conjugates (a sum and difference of the same terms).

Multiplying Three Binomials

Some expressions contain three or more polynomial factors.

The safest method is to multiply two factors first, simplify, and then multiply the result by the remaining factor.

Worked Example

Expand:

(x + 1)(x + 2)(x + 3)

First multiply:

(x + 1)(x + 2)

= x² + 3x + 2

Now multiply by the third factor:

(x² + 3x + 2)(x + 3)

Distribute x:

x³ + 3x² + 2x

Distribute 3:

3x² + 9x + 6

Combine:

x³ + 6x² + 11x + 6

Another Example

Expand:

(x − 2)(x + 2)(x + 5)

Recognise the difference of squares:

(x − 2)(x + 2) = x² − 4

Now multiply:

(x² − 4)(x + 5)

= x³ + 5x² − 4x − 20

Final answer: x³ + 5x² − 4x − 20

Recognising a special product makes this calculation shorter.

Combining Multiplication, Addition, and Subtraction

Some algebraic expressions contain several polynomial operations.

Follow the order of operations:

1. Simplify inside grouping symbols where possible.

2. Perform multiplication using distribution.

3. Remove parentheses carefully.

4. Combine like terms.

5. Write the result in standard form.

Worked Example

Simplify:

(x + 3)(x + 4) + (2x − 5)

Multiply the binomials:

(x + 3)(x + 4) = x² + 7x + 12

Substitute:

x² + 7x + 12 + 2x − 5

Combine like terms:

x² + 9x + 7

Worked Example with Subtraction

Simplify:

(2x + 1)(x − 3) − (x² − 4x + 5)

First multiply:

(2x + 1)(x − 3)

= 2x² − 6x + x − 3

= 2x² − 5x − 3

Now subtract the second polynomial:

2x² − 5x − 3 − (x² − 4x + 5)

Distribute the negative sign:

2x² − 5x − 3 − x² + 4x − 5

Combine:

x² − x − 8

Polynomial Products with Multiple Variables

The distributive property also works when polynomials contain several variables.

Worked Example

Expand:

(2a + 3b)(a + 4b)

Multiply:

2a × a = 2a²

2a × 4b = 8ab

3b × a = 3ab

3b × 4b = 12b²

Combine:

2a² + 8ab + 3ab + 12b²

= 2a² + 11ab + 12b²

Worked Example

Expand:

(x + y)²

Apply the square-of-a-sum identity:

(x + y)² = x² + 2xy + y²

The terms x², xy, and y² are not like terms, so they cannot be combined further.

Verifying Polynomial Multiplication

One way to check an expanded polynomial is to substitute the same value into the original and expanded expressions.

Worked Example

Verify:

(x + 2)(x + 5) = x² + 7x + 10

Choose x = 3.

Original expression:

(3 + 2)(3 + 5)

= 5 × 8

= 40

Expanded expression:

3² + 7(3) + 10

= 9 + 21 + 10

= 40

Both give 40.

This is a useful check, although agreement at one value does not prove equivalence for every value.

A stronger algebraic verification is to reverse the expansion using the distributive property or factoring.

Real-World Application: Finding the Area of a Rectangle

Suppose a rectangular garden has:

Length = x + 7 metres

Width = x + 3 metres

The area is:

A = Length × Width

Substitute:

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

Expand:

A = x² + 3x + 7x + 21

Combine:

A = x² + 10x + 21

If x = 5 metres:

A = 5² + 10(5) + 21

= 25 + 50 + 21

A = 96 m²

Real-World Application: Calculating Revenue

A company sells a product.

Suppose the number of units sold depends on price according to:

q = 100 − 2p

where p is the price in dollars.

Revenue is:

R = Price × Quantity

Therefore:

R = p(100 − 2p)

Distribute:

R = 100p − 2p²

This polynomial models the company's revenue.

Example

If the price is $20:

R = 100(20) − 2(20²)

= 2000 − 800

R = $1,200

Polynomial multiplication helps construct models used in business and economics.

Real-World Application: Expanding a Square Garden

Suppose a square garden has side length x metres.

Its original area is:

A₁ = x²

The garden is expanded by adding 3 metres to each side.

Its new side length is:

x + 3

The new area is:

A₂ = (x + 3)²

Expand:

A₂ = x² + 6x + 9

The increase in area is:

A₂ − A₁

= x² + 6x + 9 − x²

= 6x + 9

Interpretation

The increase in area depends on the original side length.

If x = 10 metres:

Increase = 6(10) + 9

= 69 m²

This demonstrates a practical use of the square-of-a-sum identity.

Choosing an Efficient Multiplication Strategy

The distributive property always works. Special products simply provide faster methods in particular situations.

Common Mistakes

Adding exponents incorrectly

Incorrect:

x² × x³ = x⁶

Correct:

x² × x³ = x⁵

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

Multiplying only the first terms

Incorrect:

(x + 2)(x + 3) = x² + 6

Correct:

(x + 2)(x + 3) = x² + 5x + 6

Every term must be multiplied.

Forgetting negative signs

Incorrect:

(x − 4)(x − 2) = x² − 6x − 8

Correct:

(x − 4)(x − 2) = x² − 6x + 8

Squaring terms separately

Incorrect:

(x + 5)² = x² + 25

Correct:

(x + 5)² = x² + 10x + 25

Confusing addition and multiplication of like terms

Addition:

3x² + 4x² = 7x²

Multiplication:

3x² × 4x² = 12x⁴

Combining unlike terms

Incorrect:

3x² + 5x = 8x³

Correct:

3x² + 5x cannot be combined further.

Forgetting to distribute a negative coefficient

Incorrect:

−2(x − 3) = −2x − 6

Correct:

−2(x − 3) = −2x + 6

Applying FOIL to expressions that are not two binomials

FOIL is not sufficient for multiplying a binomial by a trinomial. Use the full distributive property or a multiplication grid.

Failing to simplify after expanding

An expression such as:

x² + 3x + 4x + 12

should be simplified to:

x² + 7x + 12

Key Terms

• Polynomial: Algebraic expression consisting of terms with nonnegative integer powers of variables.

• Monomial: Polynomial containing one term.

• Binomial: Polynomial containing two terms.

• Trinomial: Polynomial containing three terms.

• Coefficient: Numerical factor multiplying a variable.

• Exponent: Number indicating the power of a variable.

• Product: Result of multiplication.

• Factor: Expression or number being multiplied.

• Distributive Property: Property allowing multiplication across terms in parentheses.

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

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

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

• FOIL: First, Outer, Inner, Last; a method for multiplying two binomials.

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

• Grid Method: Organised table for multiplying polynomial terms.

• Special Product: Polynomial product following a commonly recognised identity.

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

• Difference of Squares: Expression of the form a² − b².

• Conjugate Binomials: Binomials such as a + b and a − b.

• Standard Form: Polynomial written in descending order of powers.

• Degree: Highest exponent of a nonzero polynomial in one variable.

• Equivalent Expressions: Expressions that produce the same value for every permitted input.

• Verification: Process of checking whether a mathematical result is correct.

Key Takeaways

• Multiplying polynomials requires the distributive property.

• When multiplying monomials, multiply coefficients and add exponents of matching variables.

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

• Two binomials produce four individual products before simplification.

• FOIL is a memory aid for multiplying two binomials.

• Area models and multiplication grids can help organise polynomial multiplication.

• Binomial–trinomial multiplication produces six individual products before simplification.

• Trinomial–trinomial multiplication produces nine individual products before simplification.

• Like terms must be combined after expanding.

• The square of a sum is a² + 2ab + b².

• The square of a difference is a² − 2ab + b².

• The product of conjugate binomials is a² − b².

• Special product identities make some calculations faster.

• Expressions involving several operations should be expanded before combining like terms.

• Substitution can help check expanded expressions.

• Polynomial multiplication is useful for geometry, science, finance, and mathematical modelling.

Check Your Understanding

Multiplying Monomials

1. 3x × 4x

2. 5x² × 2x³

3. −4a² × 3a

4. 2x²y × 5xy³

Using the Distributive Property

5. 3(x + 7)

6. 4x(2x − 5)

7. −2x(3x² − 4x + 6)

8. 5a(2a² + 3a − 4)

Multiplying Binomials

9. (x + 3)(x + 5)

10. (x − 4)(x + 7)

11. (x − 6)(x − 2)

12. (2x + 3)(3x + 4)

Multiplying Larger Polynomials

13. (x + 2)(x² + 3x + 1)

14. (2x − 1)(x² + 4x + 3)

15. (x + 1)(x + 2)(x + 4)

Special Products

16. (x + 6)²

17. (x − 5)²

18. (x + 7)(x − 7)

19. (3x + 2)²

20. (4x − 3)(4x + 3)

Answer Key

Extended Challenge

Expand and simplify each expression. Show all necessary steps.

Monomial and Polynomial Products

1. −3x²(4x³ − 2x + 5)

2. 5a²b(2ab² − 3a²b + 4)

3. −2xy(3x²y − 4xy² + 5)

4. 4x³(2x² − 3x + 7)

5. −5a(3a³ − 2a² + 6a − 1)

Binomial Products

6. (3x + 5)(2x − 7)

7. (4x − 3)(5x + 2)

8. (2a + 3b)(4a − 5b)

9. (x² + 3)(x² − 5)

10. (3x² − 2x)(2x² + 5x)

Binomial and Trinomial Products

11. (x + 4)(x² − 3x + 2)

12. (2x − 5)(3x² + 2x − 4)

13. (x² + 2x + 1)(x − 3)

14. (3x + 2)(2x² − 5x + 6)

15. (x² − x + 2)(x² + 3x − 4)

Special Products

16. (2x + 7)²

17. (5x − 3)²

18. (4x + 9)(4x − 9)

19. (3a + 2b)²

20. (5x² − 2y)(5x² + 2y)

Multiple Operations

21. (x + 2)(x + 3) + (x − 4)(x + 5)

22. (2x − 3)² − (x + 4)²

23. (x + 1)(x − 1)(x + 3)

24. 2(x + 3)(x − 4) − 3(x − 2)

25. (x² + 2x + 3)(x² − x + 4)

Final Challenge: Designing a Community Garden

A community is designing a rectangular garden.

The garden has the following dimensions:

Length = 2x + 5 metres

Width = x + 3 metres

A rectangular storage area inside the garden has dimensions:

Length = x + 1 metres

Width = x − 1 metres

Assume x > 1 and that the storage area fits inside the garden.

Complete the investigation.

1. Write an expression for the total area of the garden.

2. Expand the expression using the distributive property.

3. Simplify the result.

4. Write an expression for the area of the storage area.

5. Recognise whether the storage area expression is a special product.

6. Expand and simplify the storage area expression.

7. Write an expression for the remaining garden area.

8. Subtract the storage area from the total area.

9. Simplify the resulting polynomial.

10. Evaluate the total garden area when x = 6.

11. Evaluate the storage area when x = 6.

12. Calculate the remaining garden area when x = 6.

13. Verify your results using the original dimension expressions.

14. Explain why the area expressions contain squared terms.

15. Explain how the distributive property helps calculate areas.

16. Describe why combining like terms is necessary.

17. Identify the degree of each simplified area polynomial.

18. Explain how changing x affects the garden dimensions.

19. Draw a labelled diagram of the garden and storage area.

20. Present a complete mathematical solution, including units and a written conclusion.

Further Investigation: Discovering a Pattern

Consider the following products:

(x + 1)(x + 2)

(x + 2)(x + 3)

(x + 3)(x + 4)

(x + 4)(x + 5)

Expand each expression.

Compare the coefficients of x², x, and the constant terms.

Then investigate the general product:

(x + n)(x + n + 1)

where n is a positive integer.

Show that:

(x + n)(x + n + 1) = x² + (2n + 1)x + n(n + 1)

Explain how the distributive property produces this pattern and describe how it can be used to predict polynomial products.