Expanding Algebraic Expressions

TL;DR

Expanding algebraic expressions means getting rid of parentheses by multiplying everything inside them by what's outside. It's essentially applying the distributive property multiple times. This skill is crucial for solving equations and simplifying more complex algebraic problems.

1. The Mental Model

Think of expansion like distributing mail to everyone in a house. Whatever is outside the parentheses needs to be "delivered" (multiplied) to every single term inside the parentheses. No one gets left out!

2. The Core Material

When you expand an algebraic expression, you're using the distributive property. This property states that $a(b + c) = ab + ac$. You multiply the term outside the parentheses by each term inside.

2.1 Expanding a Single Term with Parentheses

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This is the most basic form. You have one term (a number or a variable) multiplied by an expression in parentheses.

Example: $3(x + 5)$

Here, you multiply 3 by $x$ AND by 5:
$3 \times x = 3x$
$3 \times 5 = 15$
So, $3(x + 5) = 3x + 15$

Example: $-2(y - 4)$

Remember to pay attention to signs!
$-2 \times y = -2y$
$-2 \times -4 = +8$ (a negative times a negative is a positive)
So, $-2(y - 4) = -2y + 8$

2.2 Expanding Two Binomials (FOIL Method)

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When you have two sets of parentheses, like $(a+b)(c+d)$, you need to make sure each term in the first set is multiplied by each term in the second set. A common way to remember this is the FOIL method:

• First: Multiply the First terms in each binomial.
• Outer: Multiply the Outer terms.
• Inner: Multiply the Inner terms.
• Last: Multiply the Last terms.

Then, you combine any like terms.

graph TD
    Start["Start with (a + b)(c + d)"] --> F[("First: a * c")]
    F --> O[("Outer: a * d")]
    O --> I[("Inner: b * c")]
    I --> L[("Last: b * d")]
    L --> Combine["Add all results: ac + ad + bc + bd"]
    Combine --> End["Simplify by combining like terms"]

Example: $(x + 2)(x + 3)$

• First: $x \times x = x^2$
• Outer: $x \times 3 = 3x$
• Inner: $2 \times x = 2x$
• Last: $2 \times 3 = 6$

Add them up: $x^2 + 3x + 2x + 6$
Combine like terms ($3x$ and $2x$): $x^2 + 5x + 6$

Example: $(2x - 1)(x + 4)$

• First: $2x \times x = 2x^2$
• Outer: $2x \times 4 = 8x$
• Inner: $-1 \times x = -x$ (remember the negative sign!)
• Last: $-1 \times 4 = -4$

Add them up: $2x^2 + 8x - x - 4$
Combine like terms ($8x$ and $-x$): $2x^2 + 7x - 4$

2.3 Expanding a Binomial Squared

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When you see something like $(x+y)^2$, it's important to remember that this means $(x+y)(x+y)$. Don't just square each term individually! You'll use the FOIL method here too.

Example: $(x - 5)^2$

Rewrite it as: $(x - 5)(x - 5)$

• First: $x \times x = x^2$
• Outer: $x \times -5 = -5x$
• Inner: $-5 \times x = -5x$
• Last: $-5 \times -5 = +25$

Add them up: $x^2 - 5x - 5x + 25$
Combine like terms: $x^2 - 10x + 25$

3. Worked Example

Let's expand the expression: $(3x + 2)(x - 7)$

1. Identify the two binomials: We have $(3x + 2)$ and $(x - 7)$.
2. Apply FOIL:
   • First: $3x \times x = 3x^2$
   • Outer: $3x \times -7 = -21x$
   • Inner: $2 \times x = 2x$
   • Last: $2 \times -7 = -14$
3. Write out the sum of these terms: $3x^2 - 21x + 2x - 14$
4. Combine like terms: The terms $-21x$ and $2x$ are like terms.
   $-21x + 2x = -19x$
5. Final Expanded Expression: $3x^2 - 19x - 14$

4. Key Takeaways

• Expanding means removing parentheses by multiplying terms.
• The distributive property is the foundation: $a(b+c) = ab + ac$.
• For two binomials, the FOIL method (First, Outer, Inner, Last) helps ensure you multiply every term.
• Always combine any like terms after multiplying to simplify the expression.
• Be careful with negative signs; they affect the sign of the product.
• $(a+b)^2$ is NOT $a^2+b^2$; it means $(a+b)(a+b)$ and requires FOIL.

Common Mistakes to Avoid:
- Forgetting to multiply every term inside the parentheses by the term outside.
- Making sign errors when multiplying negative numbers.
- Not combining like terms after expanding.
- Incorrectly thinking $(a+b)^2 = a^2+b^2$.

5. Now Try It

Expand the expression $(y - 4)(2y + 5)$. Take about 15 minutes. Success looks like you've correctly multiplied all terms using FOIL and combined any resulting like terms into a simplified quadratic expression (something like $ay^2 + by + c$).