Algebraic Expressions, Operations, and Factoring Study Notes

Translating Verbal Phrases to Algebraic Expressions: Algebraic expressions use symbols to represent relationships between numbers and variables.
- Example: The expression representing "five more than twice a number" is formulated by identifying "twice a number" as $2x$ and adding five, resulting in $2x + 5$ .
- Available options for this phrase typically include:
  - A. $2x - 5$
  - B. $5x + 2$
  - C. $2x + 5$
  - D. $5(x + 2)$
Constant Terms: A constant term in an expression is a number that does not change because it is not attached to a variable.
- In a list of terms such as $5x$ , $-3x$ , $12$ , and $x$ , the constant term is $12$ .
Like Terms: These are terms that contain the exact same variables raised to the exact same powers.
- Examples of like terms: $3a^2$ and $-7a^2$ .
- Non-examples (Unlike terms):
  - $4x$ and $4x^2$ (different exponents)
  - $5y$ and $5z$ (different variables)
  - $2m^2$ and $8m$ (different exponents)

Evaluation of Expressions: To find the value of an expression, substitute a specific numerical value for the variable and perform the arithmetic operations.
- Example: To evaluate $3x + 7$ when $x = 4$ :
  - $3(4) + 7 = 12 + 7 = 19$
- Evaluation options often provided:
  - A. $15$
  - B. $19$
  - C. $21$
  - D. $283$
Simplifying by Combining Like Terms: This process involves adding or subtracting the coefficients of terms with identical variable parts.
- Example 1: Simplify $7x + 5x - 3$
  - Combine $7x$ and $5x$ to get $12x$ .
  - The simplified expression is $12x - 3$ .
- Example 2: Simplify $9a^2 - 4a^2 + 3a$
  - Combine the squared terms: $9a^2 - 4a^2 = 5a^2$ .
  - The simplified expression is $5a^2 + 3a$ .
- Example 3: Simplify $5m + 3n - 2m + 7n$
  - Combine $m$ terms: $5m - 2m = 3m$ .
  - Combine $n$ terms: $3n + 7n = 10n$ .
  - The simplified expression is $3m + 10n$ .

Linear Earnings Expressions: Variable rates can be modeled using algebraic expressions where each variable represents a different type of labor or unit of time.
- Scenario: A student earns \unicode{x20B1}30 for every tutoring hour $t$ and \unicode{x20B1}20 for every chore hour $c$ .
- The total earnings expression is the sum of the products of each rate and its respective time variable: $30t + 20c$ .

Polynomial Identification: A polynomial is an algebraic expression where the variables have non-negative integer exponents.
- A valid polynomial example is $4x^2 - 3x + 7$ .
- Expressions that are NOT polynomials include:
  - $\frac{3}{x} + 2$ (variable in the denominator implies a negative exponent)
  - $x^{-2} + 5$ (contains a negative exponent)
  - $\sqrt{x} + 1$ (radical notation implies a fractional exponent)
Subtraction of Polynomials: When subtracting one polynomial from another, distribute the negative sign to every term in the polynomial being subtracted before combining like terms.
- Procedure: Subtracting $4x^2 - 3x + 2$ from $7x^2 + x - 5$ .
- Align like terms: $(7x^2 - 4x^2) + (x - (-3x)) + (-5 - 2)$ .
- Calculated result: $3x^2 + 4x - 7$ .

Product of Monomials: Use the product rule for exponents, which states that $x^m \times x^n = x^{m+n}$ , and multiply coefficients normally.
- Multiplying $3x$ and $4x^2$ yields $12x^3$ .
- Simplifying $2a^3(5a^2)$ yields $10a^5$ .
- Multiplying $4x^2$ and $3y^3$ yields $12x^2y^3$ .
Expanding Expressions (Distribution): Distribute the monomial multiplier to each term inside the parentheses.
- Expanding $3x(x + 4)$ : $3x(x) + 3x(4) = 3x^2 + 12x$ .
- Expanding $2x(x^2 + x + 3)$ : $2x(x^2) + 2x(x) + 2x(3) = 2x^3 + 2x^2 + 6x$ .
Binomial Multiplication (FOIL Method): The FOIL method (First, Outer, Inner, Last) is used to multiply two binomials.
- Expanding $(x+3)(x+5)$ : $x^2 + 5x + 3x + 15 = x^2 + 8x + 15$ .
- Expanding $(x-2)(x+6)$ : $x^2 + 6x - 2x - 12 = x^2 + 4x - 12$ .
- Expanding $(2x+3)(x-4)$ : $2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12$ .
Division of Monomials: Use the quotient rule for exponents, which states that $\frac{x^m}{x^n} = x^{m-n}$ , and divide coefficients normally.
- Simplifying $\frac{18y^6}{3y^2}$ : $\frac{18}{3}y^{6-2} = 6y^4$ .
- Simplifying $\frac{20a^5b^3}{5a^2b}$ : $\frac{20}{5}a^{5-2}b^{3-1} = 4a^3b^2$ .

Square of a Binomial Formula: The expansion follows the pattern $(a + b)^2 = a^2 + 2ab + b^2$ .
- Expanding $(x + 6)^2$ : $x^2 + 2(x)(6) + 6^2 = x^2 + 12x + 36$ .
- Expanding $(3a - 2)^2$ : $(3a)^2 - 2(3a)(2) + (-2)^2 = 9a^2 - 12a + 4$ .
Difference of Two Squares: The product of a sum and a difference of the same two terms yields the square of the first term minus the square of the second term: $(a + b)(a - b) = a^2 - b^2$ .
- Simplifying $(x + 7)(x - 7)$ : $x^2 - 49$ .
- Factoring $x^2 - 25$ : $(x + 5)(x - 5)$ .
Factoring Approaches:
- Greatest Common Monomial Factor (GCMF): Identify the largest factor shared by all terms.
  - Example: For $8x^3 + 12x^2$ , the GCMF is $4x^2$ .
  - Factored form: $4x^2(2x + 3)$ .
- Perfect Square Trinomials: Expressions of the form $a^2 + 2ab + b^2$ factor into $(a + b)^2$ .
  - Example: $x^2 + 10x + 25$ factors into $(x + 5)^2$ .
- Difference of Two Squares Factorization: $9a^2 - 16$ is factored as $(3a + 4)(3a - 4)$ .
- Sum of Two Cubes: The formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ .
  - Factoring $x^3 + 8$ where $a = x$ and $b = 2$ : $(x + 2)(x^2 - 2x + 4)$ .
- Factoring by Grouping: Group terms into pairs and factor out the common binomial factor.
  - Factoring $x^3 + 3x^2 + 2x + 6$ :
    - Group: $(x^3 + 3x^2) + (2x + 6)$ .
    - Factor each group: $x^2(x + 3) + 2(x + 3)$ .
    - Complete factor: $(x + 3)(x^2 + 2)$ .