Simplifying Expressions

Simplifying algebraic expressions makes them shorter and easier to work with. You'll combine like terms, use the distributive property, and clean up expressions to their simplest form!

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What Does "Simplify" Mean?

To simplify an expression means to make it as short and clean as possible by:

• Combining like terms
• Using the distributive property
• Removing parentheses
• Writing in standard form

Example:

• Before: 3x + 2x + 5 - 2
• After: 5x + 3

Both expressions are equal, but the simplified version is cleaner!

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Like Terms

Like terms have the SAME variable raised to the SAME power.

Like Terms (can combine):

• 3x and 5x (both have x)
• -2y and 7y (both have y)
• 4x² and x² (both have x²)
• 6 and -3 (both are constants)

Unlike Terms (CANNOT combine):

• 3x and 5y (different variables)
• 4x and 4x² (different exponents)
• 2xy and 3x (different variables)

Why it matters: You can only combine like terms!

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Combining Like Terms

Add or subtract the coefficients, keep the variable part the same.

Example 1: Simple Addition

Simplify: 4x + 7x

Solution:

• Both terms have x
• Add coefficients: 4 + 7 = 11
• Keep the variable: x
• Answer: 11x

Think: 4 apples + 7 apples = 11 apples

Example 2: With Subtraction

Simplify: 9y - 3y

Solution:

• Both terms have y
• Subtract coefficients: 9 - 3 = 6
• Answer: 6y

Example 3: Multiple Terms

Simplify: 5x + 3x - 2x

Solution:

• All terms have x
• Combine: 5 + 3 - 2 = 6
• Answer: 6x

Example 4: With Constants

Simplify: 3x + 7 + 2x - 4

Solution: Step 1: Group like terms

• x terms: 3x + 2x
• Constants: 7 - 4

Step 2: Combine each group

• 3x + 2x = 5x
• 7 - 4 = 3

Step 3: Write final answer

• Answer: 5x + 3

Example 5: Negative Coefficients

Simplify: 8a - 5a + 3a

Solution:

• All have variable a
• Combine: 8 - 5 + 3 = 6
• Answer: 6a

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The Distributive Property

Formula: a(b + c) = ab + ac

Multiply the number outside the parentheses by EACH term inside.

Example 1: Basic Distribution

Simplify: 3(x + 4)

Solution:

• 3 × x = 3x
• 3 × 4 = 12
• Answer: 3x + 12

Example 2: Negative Distribution

Simplify: -2(y - 5)

Solution:

• -2 × y = -2y
• -2 × (-5) = +10
• Answer: -2y + 10

Important: Distribute the negative sign too!

Example 3: Variable Outside

Simplify: x(3 + 5)

Solution:

• x × 3 = 3x
• x × 5 = 5x
• Answer: 3x + 5x = 8x

Example 4: Distribution with Three Terms

Simplify: 4(2x - 3 + y)

Solution:

• 4 × 2x = 8x
• 4 × (-3) = -12
• 4 × y = 4y
• Answer: 8x - 12 + 4y

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Combining Distribution and Like Terms

Many problems require both steps!

Example 1: Distribute Then Combine

Simplify: 2(x + 3) + 5x

Step 1: Distribute

• 2(x + 3) = 2x + 6

Step 2: Rewrite

• 2x + 6 + 5x

Step 3: Combine like terms

• x terms: 2x + 5x = 7x
• Constants: 6

Answer: 7x + 6

Example 2: Multiple Distributions

Simplify: 3(x + 2) + 4(x - 1)

Step 1: Distribute first parentheses

• 3(x + 2) = 3x + 6

Step 2: Distribute second parentheses

• 4(x - 1) = 4x - 4

Step 3: Rewrite

• 3x + 6 + 4x - 4

Step 4: Combine like terms

• x terms: 3x + 4x = 7x
• Constants: 6 - 4 = 2

Answer: 7x + 2

Example 3: With Negative Distribution

Simplify: 5(2y + 1) - 3(y - 4)

Step 1: Distribute

• 5(2y + 1) = 10y + 5
• -3(y - 4) = -3y + 12 (watch the signs!)

Step 2: Rewrite

• 10y + 5 - 3y + 12

Step 3: Combine

• y terms: 10y - 3y = 7y
• Constants: 5 + 12 = 17

Answer: 7y + 17

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Removing Parentheses

Positive Sign Before Parentheses

Just remove the parentheses - nothing changes!

Example: 3x + (2x + 5) = 3x + 2x + 5 = 5x + 5

Negative Sign Before Parentheses

Change the sign of EVERY term inside!

Example 1: 4x - (2x + 3)

• Remove parentheses: 4x - 2x - 3
• Combine: 2x - 3

Example 2: 7y - (3y - 5)

• Remove parentheses: 7y - 3y + 5
• Combine: 4y + 5

Think of it as: -(2x + 3) = -1(2x + 3) = -2x - 3

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Simplifying with Multiple Variables

Combine terms with the same variable!

Example 1: Two Variables

Simplify: 3x + 2y + 5x - y

Solution:

• x terms: 3x + 5x = 8x
• y terms: 2y - y = y
• Answer: 8x + y

Example 2: Three Variables

Simplify: 4a + 3b - 2a + 5c - b

Solution:

• a terms: 4a - 2a = 2a
• b terms: 3b - b = 2b
• c terms: 5c
• Answer: 2a + 2b + 5c

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Simplifying with Exponents

Remember: Only combine terms with the same variable AND same exponent!

Example 1: Same Exponents

Simplify: 5x² + 3x² - 2x²

Solution:

• All have x²
• Combine: 5 + 3 - 2 = 6
• Answer: 6x²

Example 2: Different Exponents

Simplify: 4x² + 3x + 2x² - x

Solution:

• x² terms: 4x² + 2x² = 6x²
• x terms: 3x - x = 2x
• Answer: 6x² + 2x

Cannot combine x² and x - they're unlike terms!

Example 3: Mixed Variables and Exponents

Simplify: 2xy + 3x + xy - 5x

Solution:

• xy terms: 2xy + xy = 3xy
• x terms: 3x - 5x = -2x
• Answer: 3xy - 2x

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Order of Operations in Simplifying

Follow PEMDAS when simplifying!

Example: Simplify: 2(3x + 4) - 5 + 3x

Step 1: Parentheses/Distribution first

• 2(3x + 4) = 6x + 8

Step 2: Rewrite

• 6x + 8 - 5 + 3x

Step 3: Combine like terms

• x terms: 6x + 3x = 9x
• Constants: 8 - 5 = 3

Answer: 9x + 3

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Real-World Applications

Perimeter Problems

Problem: A rectangle has length (3x + 2) and width (x + 5). Write a simplified expression for the perimeter.

Solution: Perimeter = 2(length) + 2(width) P = 2(3x + 2) + 2(x + 5) P = 6x + 4 + 2x + 10 P = 8x + 14

Answer: Perimeter = 8x + 14

Shopping

Problem: You buy 3 shirts at $x each and 2 pairs of pants at$y each. Then you return 1 shirt. Write a simplified expression for the total cost.

Solution: Total = 3x + 2y - x Total = 2x + 2y

Answer: 2x + 2y

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Common Mistakes to Avoid

❌ Mistake 1: Combining unlike terms

• Wrong: 3x + 4y = 7xy
• Right: 3x + 4y (cannot combine - different variables!)

❌ Mistake 2: Forgetting to distribute negative

• Wrong: 5 - (x + 3) = 5 - x + 3 = 8 - x
• Right: 5 - (x + 3) = 5 - x - 3 = 2 - x

❌ Mistake 3: Distributing to only one term

• Wrong: 2(x + 3) = 2x + 3
• Right: 2(x + 3) = 2x + 6

❌ Mistake 4: Combining different exponents

• Wrong: 2x² + 3x = 5x³
• Right: 2x² + 3x (cannot combine!)

❌ Mistake 5: Sign errors

• Wrong: 4x - 2x = 2x or -2x? (confusion)
• Right: 4x - 2x = 2x (4 minus 2 is positive 2)

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Practice Strategy

Step 1: Look for parentheses

• Distribute first!
• Remove negative signs carefully

Step 2: Identify like terms

• Circle or underline terms with the same variable/exponent

Step 3: Combine

• Add/subtract coefficients
• Keep variable parts the same

Step 4: Write in standard form

• Usually highest exponent first
• Then lower exponents
• Then constants

Step 5: Check

• Can you combine anything else?
• Are all like terms together?

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Standard Form

Standard form for polynomials: Write terms in descending order of exponents.

Example: Simplify and write in standard form: 5 + 3x² - 2x + x²

Step 1: Combine like terms

• x² terms: 3x² + x² = 4x²
• x terms: -2x
• Constants: 5

Step 2: Write in standard form (highest exponent first)

• Answer: 4x² - 2x + 5

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Quick Tips

Tip 1: Like terms are "friends" - they can combine!

• 3x and 5x are friends → 8x
• 3x and 5y are NOT friends → stay separate

Tip 2: Distribute carefully with negatives

• -(3x - 2) means multiply EVERYTHING by -1
• Result: -3x + 2

Tip 3: Use different colors for different variables

• Circle all x terms in blue
• Circle all y terms in red
• Makes it easy to see what combines!

Tip 4: Check by substituting a number

• Original: 2(x + 3) + 4x
• Simplified: 6x + 6
• Test with x = 1: 2(1+3) + 4(1) = 2(4) + 4 = 12 ✓
• Check: 6(1) + 6 = 12 ✓

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Summary

Simplifying expressions means making them as short as possible by:

1. Using distributive property: a(b + c) = ab + ac
2. Combining like terms: Terms with same variable and exponent
3. Removing parentheses: Watch for negative signs!
4. Writing in standard form: Highest exponent first

Key Rules:

• Only like terms can combine
• Distribute to EVERY term inside parentheses
• Negative before parentheses changes ALL signs inside
• Different variables or exponents = unlike terms

Mastering simplification is essential for solving equations, factoring, and all of algebra!