Expanding expressions using properties of operations involves applying various mathematical rules and properties such as the distributive property, the associative property, and the commutative property. Below are some key properties and how they can be applied to expand expressions:
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Distributive Property:
- The distributive property states that \( a(b + c) = ab + ac \).
- This property allows you to expand expressions by distributing a term over a sum or difference.
- Example: Expand \( 3(x + 4) \): \[ 3(x + 4) = 3x + 12 \]
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Combining Like Terms:
- When you have multiple terms that are similar (like terms), you can combine them to simplify or expand the expression.
- Example: Expand \( 2x + 3x + 4 \): \[ 2x + 3x + 4 = 5x + 4 \]
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Associative Property:
- The associative property states that the way in which numbers are grouped does not change their sum or product: \( (a + b) + c = a + (b + c) \) and \( (ab)c = a(bc) \).
- This property allows you to reorganize terms for easier expansion.
- Example: Expand \( (x + 2) + (x + 3) \): \[ = x + 2 + x + 3 = 2x + 5 \]
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Commutative Property:
- The commutative property states that the order of addition or multiplication does not affect the result: \( a + b = b + a \) and \( ab = ba \).
- While primarily useful in rearranging terms, it helps in organizing terms before expansion.
- Example: Expand \( 4 + x + 2x \): \[ = 4 + x + 2x = 4 + 3x \]
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Applying Powers:
- When expressions are raised to a power, you can use the power of a product property \( (ab)^n = a^n b^n \) or the binomial theorem for expanding expressions like \( (a + b)^2 \).
- Example: Expand \( (x + 2)^2 \): \[ (x + 2)^2 = x^2 + 4x + 4 \]
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Factoring and Re-expanding:
- Sometimes it helps to factor an expression first, then expand again to see how the expression can be rewritten.
- Example: Expand \( x(x + 3) \): \[ x(x + 3) = x^2 + 3x \]
By systematically applying these properties, you can effectively expand various algebraic expressions.
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