# Mathematics

## Lesson 1: Simplifying Algebraic Expressions

### Examples:

Write the factors for each:

- 6x
^{2} - -3xy
- xy
^{3} - -x
^{3}y

### Examples:

Write the factors for each:

- 6x
^{2}

6 • x • x - -3xy

-3 • x • y - xy
^{3}

1 • x • y • y • y - -x
^{3}y

-1 • x • x • x • y

**Notice:**

- These are all 1 term expressions; terms are separated by plus or minus.
- Recall: exponent works on what's to its immediate left.
- Every term has a number factor (even if you do not see it) and a variable factor (the variable and their exponents)

**FYI:** The number factor of a term is called the coefficient or numerical coefficient.

### More Info:

**Example:** 6x^{2}

6 is the coefficient

x^{2} is the variable factor

Note: Terms with the same variable factor are __like terms__.

### Examples:

Identify the coefficient and variable factor of each term:

Term | -3xy | x^{3}y^{2} | 7a^{2}bc | -x^{3}yz^{4} |
---|---|---|---|---|

Coeficient | ||||

Variable Factor |

Separate each list into groups of like terms:

- 2x
^{2}, -3x, 4x^{3}, -x, -x^{2} - -x
^{2}y, 2xy^{2}, x^{2}y^{2}, 3xy^{2}, -2x^{2}y

### Examples:

Evaluate each expression when x = 3.

- x + 10
- 2x + 30 - x - 20

### Answers:

Evaluate each expression when x = 3.

- x + 10

(3) + 10 - 2x + 30 - x - 20

2(3) + 30 - (3) - 20

6 + 30 - 3 - 20

36 - 3 - 20

33 - 20

13

What did you notice?

### Simplify:

2x + 30 - x - 20

2x + 30 - x - 20

2x - x + 30 - 20

1x + 10

**Answer:** 1x + 10 or x + 10

**Procedure to combine Terms:**

- Look for like terms
- Combine coefficients
- Keep like term

Are you surprised?

### A Quick Look

Is 3xy = 3yx?

### A Quick Look

Is 3xy = 3yx?

3 • x • y = 3 • y • x ← commutative property holds for multiplication, so this is true!

**Numerical Example:** use x = 5 and y = -2

3xy = 3yx

3(5)(-2) = 3(-2)(5) ← 3 factors multiply, any two factors first, then the answer by the next factor.

15(-2) = -6(5)

-30 = -30 True

### Examples:

Simplify each, if possible:

- 4x
^{2}- 3x + 9 - 8x^{2}+ 1 - 4x - 3mn - 5nm + 7mn
- 8a
^{3}- 2a^{2} - -10ab + 10ab
- 7x
^{2}y - 2xy^{2}- 4xy^{2}- x^{2}y

### What Does this Mean?

2(8a)

### What Does this Mean?

2(8a)

There are two 8a's

8a + 8a (repeated addition)

16a

**So:** 2(8a) = 16a

Repeated addition is multiplication.

2(8a) is one term (no plus or minus) but it's three factors: 2 • 8 • a

2(8a) = 16a (just multiplication, multiply the numbers and keep the variable factor)

### Examples:

Find the product of each:

- -5(3x
^{2}y) - 20(-2y)
- -10(-b)

### True or False:

5(3+4) = 5(3) + 5(4)

### True or False:

5(3+4) = 5(3) + 5(4)

Order of operations: 5(7) = 15 + 20

35 = 35 True

Show this only works when terms are inside the parentheses.

### Distributive Property

Distributive Property is multiplication over terms!

You need at least two operations:

- One must be multiplication
- The other(s) must be addition and/or subtraction

**FYI:** One of the purposes of the distributive property is to be able to simplify when the terms inside the parentheses cannot be combined.

### Examples:

Simplify, use the distributive property when __necessary__:

- 5(x + 4)
- -2(3a
^{2}- 4) - 5(x + 4x)
- 5(2x)(-3)
- -(3x - 2)

Do not go distributive happy!

### Putting it all Together

Simplify each:

- -3 + 7(x + 2)
- 4(7x + 2) + 5x
- 10 + 3(x
^{2}- 2)

### One Step Further

Simplfy:

5(-2y) - 3(4y - 7) - 6 + 2(y - 1) - (7y - 10y)