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Mathematics

Lesson 1: Simplifying Algebraic Expressions

Examples:

Write the factors for each:

  1. 6x2
  2. -3xy
  3. xy3
  4. -x3y

Examples:

Write the factors for each:

  1. 6x2
    6 • x • x
  2. -3xy
    -3 • x • y
  3. xy3
    1 • x • y • y • y
  4. -x3y
    -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: 6x2

6 is the coefficient

x2 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-3xyx3y27a2bc-x3yz4
Coeficient
Variable Factor

Separate each list into groups of like terms:

  1. 2x2, -3x, 4x3, -x, -x2
  2. -x2y, 2xy2, x2y2, 3xy2, -2x2y

Examples:

Evaluate each expression when x = 3.

  1. x + 10
  2. 2x + 30 - x - 20

Answers:

Evaluate each expression when x = 3.

  1. x + 10
    (3) + 10
  2. 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:

  1. Look for like terms
  2. Combine coefficients
  3. 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:

  1. 4x2 - 3x + 9 - 8x2 + 1 - 4x
  2. 3mn - 5nm + 7mn
  3. 8a3 - 2a2
  4. -10ab + 10ab
  5. 7x2y - 2xy2 - 4xy2 - x2y

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:

  1. -5(3x2y)
  2. 20(-2y)
  3. -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:

  1. 5(x + 4)
  2. -2(3a2 - 4)
  3. 5(x + 4x)
  4. 5(2x)(-3)
  5. -(3x - 2)

Do not go distributive happy!

Putting it all Together

Simplify each:

  1. -3 + 7(x + 2)
  2. 4(7x + 2) + 5x
  3. 10 + 3(x2 - 2)

One Step Further

Simplfy:

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

Assessment: Vocabulary Review & Puzzle