Chapter 3 - Expressions

Chapter 3 - Algebraic Expressions

Simplifying Algebraic Expressions

Example

 $9x-2+7-x$  

Rewrite as a sum of terms:  $9x+\left(-2\right)+7+\left(-x\right)$  
Terms:  $9x,\ -2,\ 7,\ -x$  
Like Terms:  $9x\ and\ -x,\ 7\ and\ -2$  
Simplified:  $8x+5$  

Another Example

 $6n-10n=\left(6-10\right)n$  Distributive Property
 $=-4n$  Subtract

Adding and Subtracting Linear Expressions

Vocabulary

• Linear Expression - an algebraic expression where the exponent of each variable is 1.

• Linear Expressions:

   $-4x,\ 3x+5y,\ 5-\frac{1}{6}x$  

• Nonlinear Expressions:  $\frac{1}{2}x^2,\ -7x^3+x,\ x^5+1$  

Adding Expressions Example

 $\left(x-2\right)+\left(3x+8\right)$  

Rewrite the sum:  $-4y+3+11y-5$  
Use the Commutative Property of Addition:  $-4y+11y+3-5$  
Group like terms:  $\left(-4y+11y\right)+\left(3-5\right)$  
Combine like terms:  $7y-2$  

Subtracting Expressions Example

 $\left(5x+6\right)-\left(-x+6\right)$  

Add the opposite:  $\left(7y+5\right)+\left(-8y+6\right)$  
Use the Commutative Property of Addition: $7y+\left(-8y\right)+5+6$  
Group Like terms:  $\left[7y+\left(-8y\right)\right]+\left(5+6\right)$  
Combine Like Terms:  $-y+11$  

The Distributive Property

Example

 $-\frac{1}{3}\left(3n-6\right)$  

Use the distributive property:  $-\frac{1}{3}\left(3n\right)-\left(-\frac{1}{3}\right)\left(6\right)$  
Multiply:  $-n-\left(-2\right)$  
Add the opposite:  $-n+2$  

Another Example

 $-3\left(-1+2x+7\right)$  

Distributive Property:  $-3\left(-1\right)+\left(-3\right)\left(2x\right)+\left(-3\right)\left(7\right)$  
Multiply: $3+\left(-6x\right)+\left(-21\right)$  
Combine Like Terms:  $-6x-18$  

Factoring Expressions

Vocabulary

• Factoring an Expression: Writing an algebraic expression as a product of factors.

Example

 $24x-18$  Factoring Out GCF

Find GCF:  $24x:\ 1,\ 2,\ 3,\ 4,\ 6,\ 8,\ 12,\ 24,\ x$  
 $18:\ 1,\ 2,\ 3,\ 6,\ 9,\ 18$  
GCF: 6
Rewrite using GCF:  $6\left(4x\right)-6\left(3\right)$  
Distributive Property:  $6\left(4x-3\right)$  

Another Example

Factoring out a Rational number:
Factor  $\frac{1}{2}$  out of  $\frac{1}{2}x+\frac{3}{2}$  

 $\frac{1}{2}\left(x\right)++\frac{1}{2}\left(3\right)$  Rewrite the expression
 $\frac{1}{2}\left(x+3\right)$  Distributive Property

Yet Another Example

Factoring Out a negative Number: Factor -2 out of  $-4p+10$  .

Rewrite the expression:  $-2\left(2p\right)+\left(-2\right)\cdot\left(-5\right)$  
Distributive Property:  $-2\left[2p+\left(-5\right)\right]$  
Simplify:  $-2\left(2p-5\right)$