The Central Idea

The distributive property

\[ a(b+c)=ab+ac \]

is the foundation of polynomial multiplication.

When both factors contain several terms, distribution is applied more than once.

For example,

\[ (x+2)(x+5) \]

means that every term in the first factor multiplies every term in the second.

\[ x(x)+x(5)+2(x)+2(5). \]

Therefore,

\[ x^2+5x+2x+10 \]

and

\[ \boxed{(x+2)(x+5)=x^2+7x+10}. \]

Distribution Is an Equality

Distribution does not change the value of an expression. It changes only its form.

The expressions

\[ 3(x+4) \]

and

\[ 3x+12 \]

are equivalent because

\[ \boxed{3(x+4)=3x+12}. \]

Polynomial expansion works by creating equivalent expressions in which products have been rewritten as sums.

Distribution with More Than Two Terms

A factor distributes across every term inside the grouping symbols.

Consider

\[ 4x(2x^2-3x+5). \]

Multiply \(4x\) by each term:

\[ (4x)(2x^2)+(4x)(-3x)+(4x)(5). \]

Therefore,

\[ \boxed{8x^3-12x^2+20x}. \]

Distributing a Negative Factor

Negative signs must be carried through every multiplication.

For example,

\[ -3x(x^2-4x+7) \]

becomes

\[ -3x^3+12x^2-21x. \]

Thus,

\[ \boxed{-3x(x^2-4x+7)=-3x^3+12x^2-21x}. \]

Repeated Distribution

Suppose both factors contain several terms:

\[ (a+b)(c+d). \]

First distribute the first grouping:

\[ a(c+d)+b(c+d). \]

Then distribute again:

\[ ac+ad+bc+bd. \]

Therefore,

\[ \boxed{(a+b)(c+d)=ac+ad+bc+bd}. \]

This is the basic structure behind every two-binomial product.

FOIL Is a Memory Aid, Not a New Law

For two binomials, some algebra courses use the mnemonic FOIL:

For

\[ (a+b)(c+d), \]

FOIL produces

\[ ac+ad+bc+bd. \]

FOIL is simply repeated distribution written as a convenient four-step checklist.

The distributive property is the general method. FOIL applies only to the special case of two binomials.

Example: FOIL

Expand

\[ (2x+3)(x+4). \]

First:

\[ (2x)(x)=2x^2. \]

Outer:

\[ (2x)(4)=8x. \]

Inner:

\[ 3(x)=3x. \]

Last:

\[ 3(4)=12. \]

Combine:

\[ 2x^2+8x+3x+12. \]

Therefore,

\[ \boxed{(2x+3)(x+4)=2x^2+11x+12}. \]

FOIL with Negative Terms

Expand

\[ (3x-2)(x-5). \]

The four products are

\[ 3x^2,\qquad -15x,\qquad -2x,\qquad 10. \]

Combine the middle terms:

\[ \boxed{3x^2-17x+10}. \]

Sign control matters as much as multiplication.

The Area Model

Distribution can be visualized geometrically.

Suppose a rectangle has side lengths

\[ x+2 \]

and

\[ x+3. \]

Split one side into lengths \(x\) and \(2\), and the other into lengths \(x\) and \(3\).

The rectangle is divided into four smaller rectangles with areas

\[ x^2,\qquad3x,\qquad2x,\qquad6. \]

The total area is

\[ x^2+3x+2x+6 \]

or

\[ \boxed{x^2+5x+6}. \]

The area model makes the pairwise products visible.

The Box Method

The same area idea can be organized in a multiplication box.

For

\[ (x+2)(x+3), \]
\(x\) \(3\)
\(x\) \(x^2\) \(3x\)
\(2\) \(2x\) \(6\)

Add the four entries:

\[ x^2+3x+2x+6 = \boxed{x^2+5x+6}. \]

The box method is especially useful when visual organization reduces sign or omission errors.

Multiplying a Binomial by a Trinomial

FOIL is not enough when a factor has more than two terms. Use full distribution.

Expand

\[ (x+2)(x^2-3x+4). \]

Distribute \(x\):

\[ x^3-3x^2+4x. \]

Distribute \(2\):

\[ 2x^2-6x+8. \]

Combine:

\[ \boxed{x^3-x^2-2x+8}. \]

Why Special Products Exist

Certain polynomial products occur so frequently that it is useful to recognize their structure before expanding them term by term.

These are called special products.

Important examples include:

Every one of these patterns can be derived directly from distribution.

Square of a Sum

Consider

\[ (a+b)^2. \]

The exponent \(2\) means

\[ (a+b)(a+b). \]

Distribute:

\[ a^2+ab+ab+b^2. \]

Combine the middle terms:

\[ \boxed{(a+b)^2=a^2+2ab+b^2}. \]

The Middle Term in a Binomial Square

A common error is to write

\[ (a+b)^2=a^2+b^2. \]

This omits the two cross-products

\[ ab+ab=2ab. \]

The correct identity is

\[ \boxed{(a+b)^2=a^2+2ab+b^2}. \]

Example: Square of a Sum

Expand

\[ (x+5)^2. \]

Use

\[ (a+b)^2=a^2+2ab+b^2. \]

Here,

\[ a=x,\qquad b=5. \]

Therefore,

\[ x^2+2(x)(5)+25 \]

and

\[ \boxed{(x+5)^2=x^2+10x+25}. \]

Square of a Difference

Consider

\[ (a-b)^2. \]

Rewrite:

\[ (a-b)(a-b). \]

Distribute:

\[ a^2-ab-ab+b^2. \]

Combine:

\[ \boxed{(a-b)^2=a^2-2ab+b^2}. \]

Example: Square of a Difference

Expand

\[ (2x-3)^2. \]

Use

\[ (a-b)^2=a^2-2ab+b^2. \]

Here,

\[ a=2x,\qquad b=3. \]

Then

\[ (2x)^2-2(2x)(3)+3^2. \]

Therefore,

\[ \boxed{4x^2-12x+9}. \]

Perfect-Square Trinomials

The expansions

\[ a^2+2ab+b^2 \]

and

\[ a^2-2ab+b^2 \]

are called perfect-square trinomials because they are produced by squaring binomials.

Later, when factoring, we will reverse this process:

\[ a^2+2ab+b^2=(a+b)^2 \]

and

\[ a^2-2ab+b^2=(a-b)^2. \]

Conjugate Binomials

Two binomials of the form

\[ a+b \]

and

\[ a-b \]

are called conjugates.

Their product has a useful cancellation pattern:

\[ (a+b)(a-b). \]

Product of Conjugates

Expand

\[ (a+b)(a-b). \]

Distribution gives

\[ a^2-ab+ab-b^2. \]

The middle terms cancel:

\[ -ab+ab=0. \]

Therefore,

\[ \boxed{(a+b)(a-b)=a^2-b^2}. \]

Difference of Squares

The expression

\[ a^2-b^2 \]

is called a difference of squares.

It appears naturally as the product of conjugates:

\[ \boxed{a^2-b^2=(a+b)(a-b)}. \]

In this lesson we are using the identity as a multiplication pattern. Later we will reverse it as a factoring pattern.

Example: Product of Conjugates

Expand

\[ (x+7)(x-7). \]

Recognize

\[ a=x,\qquad b=7. \]

Then

\[ (x+7)(x-7)=x^2-7^2. \]

Therefore,

\[ \boxed{x^2-49}. \]

Example: Conjugates with Coefficients

Expand

\[ (3x+5)(3x-5). \]

Let

\[ a=3x,\qquad b=5. \]

Then

\[ a^2-b^2 = (3x)^2-5^2. \]

Therefore,

\[ \boxed{9x^2-25}. \]

Why the Middle Terms Disappear

In a conjugate product,

\[ (a+b)(a-b), \]

the cross-products are opposites:

\[ -ab \]

and

\[ +ab. \]

Their sum is zero.

This is why the product has only two terms:

\[ a^2-b^2. \]

Comparing the Three Core Patterns

Product Expansion
\((a+b)^2\) \(a^2+2ab+b^2\)
\((a-b)^2\) \(a^2-2ab+b^2\)
\((a+b)(a-b)\) \(a^2-b^2\)

The first two contain a doubled middle term. The conjugate product does not because the middle terms cancel.

Do Not Confuse a Square with Conjugates

Compare

\[ (x+4)^2 \]

with

\[ (x+4)(x-4). \]

The first gives

\[ x^2+8x+16. \]

The second gives

\[ x^2-16. \]

These are different patterns and should not be interchanged.

Square of a General Linear Binomial

Consider

\[ (ax+b)^2. \]

Using the square-of-a-sum pattern,

\[ (ax)^2+2(ax)b+b^2. \]

Therefore,

\[ \boxed{(ax+b)^2=a^2x^2+2abx+b^2}. \]

Square of a Linear Difference

Similarly,

\[ (ax-b)^2 \]

gives

\[ (ax)^2-2(ax)b+b^2. \]

Thus,

\[ \boxed{(ax-b)^2=a^2x^2-2abx+b^2}. \]

Conjugates of Linear Binomials

For

\[ (ax+b)(ax-b), \]

the difference-of-squares pattern gives

\[ (ax)^2-b^2. \]

Therefore,

\[ \boxed{(ax+b)(ax-b)=a^2x^2-b^2}. \]

Example: A More Complicated Binomial Square

Expand

\[ (3x^2+2y)^2. \]

Let

\[ a=3x^2,\qquad b=2y. \]

Then

\[ a^2+2ab+b^2. \]

Compute each part:

\[ (3x^2)^2=9x^4, \] \[ 2(3x^2)(2y)=12x^2y, \] \[ (2y)^2=4y^2. \]

Therefore,

\[ \boxed{9x^4+12x^2y+4y^2}. \]

Example: A More Complicated Conjugate Product

Expand

\[ (2x^3+5y^2)(2x^3-5y^2). \]

Recognize the conjugate pattern:

\[ a=2x^3,\qquad b=5y^2. \]

Then

\[ a^2-b^2. \]

Therefore,

\[ \boxed{4x^6-25y^4}. \]

Three-Factor Products

Special products can also simplify larger multiplications.

Consider

\[ (x+3)(x-3)(x+2). \]

Multiply the conjugates first:

\[ (x+3)(x-3)=x^2-9. \]

Then multiply:

\[ (x^2-9)(x+2). \]

Distribute:

\[ x^3+2x^2-9x-18. \]

Therefore,

\[ \boxed{x^3+2x^2-9x-18}. \]

Recognizing a useful pair can reduce the amount of work.

Choosing the Best Multiplication Method

Several methods may be valid for the same product.

You may use:

Good algebra is not just knowing a method. It is recognizing which method makes the structure easiest to see and least prone to error.

Pattern Recognition Versus Memorization

Memorizing

\[ (a+b)^2=a^2+2ab+b^2 \]

can save time, but the identity should remain connected to distribution.

If the pattern is forgotten, it can always be rebuilt:

\[ (a+b)^2=(a+b)(a+b). \]

This makes the method recoverable rather than fragile.

Special Products as Polynomial Identities

The equations

\[ (a+b)^2=a^2+2ab+b^2, \] \[ (a-b)^2=a^2-2ab+b^2, \]

and

\[ (a+b)(a-b)=a^2-b^2 \]

are identities.

They are true for every allowable value of the variables.

Later, polynomial identities will be studied more broadly, but these familiar examples already show how one algebraic form can be replaced by another equivalent form.

A Numerical Example

Special products also work with ordinary numbers.

Consider

\[ 103^2. \]

Write

\[ 103=100+3. \]

Then

\[ 103^2=(100+3)^2. \]

Use the square-of-a-sum identity:

\[ 100^2+2(100)(3)+3^2. \]

Therefore,

\[ 10000+600+9 = \boxed{10609}. \]

Polynomial identities can therefore become mental-arithmetic tools.

Another Numerical Shortcut

Consider

\[ 52\cdot48. \]

Write

\[ 52=50+2 \]

and

\[ 48=50-2. \]

Then

\[ 52\cdot48=(50+2)(50-2). \]

Use the difference-of-squares identity:

\[ 50^2-2^2 = 2500-4. \]

Therefore,

\[ \boxed{2496}. \]

Distributing Across Nested Expressions

Sometimes more than one layer of grouping is present.

Consider

\[ 2x\left[(x+3)(x-1)\right]. \]

First multiply the binomials:

\[ (x+3)(x-1)=x^2+2x-3. \]

Then distribute \(2x\):

\[ 2x^3+4x^2-6x. \]

Therefore,

\[ \boxed{2x(x+3)(x-1)=2x^3+4x^2-6x}. \]

Distributing a Polynomial Across a Sum

The distributive property applies to whole polynomials, not only monomials.

If

\[ P(x)=x^2+1, \]

then

\[ P(x)(Q(x)+R(x)) \]

can be written

\[ P(x)Q(x)+P(x)R(x). \]

Symbolically,

\[ \boxed{P(Q+R)=PQ+PR}. \]

This is the same distributive law already used with numbers.

Common Error: Squaring Each Term Separately

A frequent mistake is

\[ (a+b)^2=a^2+b^2. \]

The error is that squaring the whole binomial means multiplying the binomial by itself:

\[ (a+b)(a+b). \]

The cross-products create the missing middle term:

\[ \boxed{2ab}. \]

Common Error: Wrong Sign in a Squared Difference

Another common mistake is

\[ (a-b)^2=a^2-b^2. \]

But

\[ (a-b)^2=(a-b)(a-b), \]

so the middle terms are both negative:

\[ -ab-ab=-2ab. \]

The correct identity is

\[ \boxed{(a-b)^2=a^2-2ab+b^2}. \]

Common Error: Treating a Sum of Squares as a Difference of Squares

The identity

\[ a^2-b^2=(a+b)(a-b) \]

applies to a difference of squares.

It does not say

\[ a^2+b^2=(a+b)(a-b). \]

In fact,

\[ (a+b)(a-b)=a^2-b^2. \]

The sign between the square terms is essential.

Common Error: Using FOIL Beyond Binomials

FOIL describes four products because two binomials contain two terms each.

For

\[ (x+2)(x^2+3x+4), \]

there are six pairwise products, not four.

Use the general distributive rule: every term in one factor multiplies every term in the other.

Common Error: Forgetting to Square the Coefficient

Consider

\[ (3x)^2. \]

The exponent applies to the entire factor:

\[ (3x)^2=(3x)(3x). \]

Therefore,

\[ \boxed{(3x)^2=9x^2}, \]

not \(3x^2\).

Structural Checks for Special Products

Pattern recognition also provides quick error checks.

A squared binomial such as

\[ (ax+b)^2 \]

must produce:

A conjugate product such as

\[ (ax+b)(ax-b) \]

must have no middle term after simplification.

A Reliable Procedure for Distribution

  1. Identify every term in each factor.
  2. Multiply every term in one factor by every term in the other.
  3. Track signs carefully.
  4. Multiply coefficients.
  5. Add exponents on matching variable bases.
  6. Combine like terms.
  7. Write the result in standard form.

A Reliable Procedure for Special Products

  1. Inspect the structure before multiplying.
  2. Ask whether the expression matches a known pattern.
  3. If it does, identify \(a\) and \(b\) carefully.
  4. Apply the identity.
  5. Simplify powers and coefficients.
  6. Check by ordinary distribution when uncertain.

Practice 1: Distribute

Expand

\[ 5x(2x^2-3x+4). \]

Practice 2: FOIL

Expand

\[ (3x+4)(2x-5). \]

Practice 3: Binomial by Trinomial

Expand

\[ (x-2)(x^2+3x+5). \]

Practice 4: Square of a Sum

Expand

\[ (x+6)^2. \]

Practice 5: Square of a Difference

Expand

\[ (2x-5)^2. \]

Practice 6: Conjugates

Expand

\[ (4x+3)(4x-3). \]

Practice 7: Multivariable Special Product

Expand

\[ (2x^2+3y)^2. \]

Practice 8: Mental Arithmetic

Use a special-product identity to calculate

\[ 97^2. \]

Practice 9: Difference of Squares Numerically

Use a special-product identity to calculate

\[ 64\cdot56. \]

Practice 10: Identify the Error

A student writes

\[ (x-4)^2=x^2-16. \]

Explain the mistake and give the correct expansion.

Practice Answers

Practice 1

\[ 5x(2x^2-3x+4) = \boxed{10x^3-15x^2+20x}. \]

Practice 2

\[ (3x+4)(2x-5) \] \[ = 6x^2-15x+8x-20 \] \[ = \boxed{6x^2-7x-20}. \]

Practice 3

\[ (x-2)(x^2+3x+5) \] \[ = x^3+3x^2+5x-2x^2-6x-10 \] \[ = \boxed{x^3+x^2-x-10}. \]

Practice 4

\[ (x+6)^2 = x^2+12x+36. \]

Therefore,

\[ \boxed{x^2+12x+36}. \]

Practice 5

\[ (2x-5)^2 = (2x)^2-2(2x)(5)+25 \] \[ = \boxed{4x^2-20x+25}. \]

Practice 6

\[ (4x+3)(4x-3) = (4x)^2-3^2 \] \[ = \boxed{16x^2-9}. \]

Practice 7

\[ (2x^2+3y)^2 \] \[ = (2x^2)^2+2(2x^2)(3y)+(3y)^2 \] \[ = \boxed{4x^4+12x^2y+9y^2}. \]

Practice 8

Write

\[ 97=100-3. \]

Then

\[ 97^2=(100-3)^2 \] \[ = 10000-600+9 \] \[ = \boxed{9409}. \]

Practice 9

Write

\[ 64=60+4 \]

and

\[ 56=60-4. \]

Then

\[ 64\cdot56=(60+4)(60-4) \] \[ = 60^2-4^2 \] \[ = 3600-16 \] \[ = \boxed{3584}. \]

Practice 10

The student confused a squared difference with a product of conjugates.

\[ (x-4)^2=(x-4)(x-4). \]

Therefore,

\[ x^2-4x-4x+16 \] \[ = \boxed{x^2-8x+16}. \]

What to Remember

Polynomial multiplication is repeated distribution.

For two binomials,

\[ (a+b)(c+d)=ac+ad+bc+bd. \]

FOIL is only a mnemonic for this two-binomial case.

The three most important special products are

\[ \boxed{(a+b)^2=a^2+2ab+b^2}, \] \[ \boxed{(a-b)^2=a^2-2ab+b^2}, \]

and

\[ \boxed{(a+b)(a-b)=a^2-b^2}. \]

These patterns are polynomial identities. They can be used both as efficient multiplication tools and, later, in reverse as factoring tools.

Where We Go Next

Expansion rewrites products as sums. Factoring reverses that direction by rewriting sums as products.

The next lesson, Common Factors and Factoring by Grouping, begins that reverse process by identifying shared factors and organizing terms into useful groups.