Form II Chapter II: Factorization and Algebraic Fractions

• Definition of Algebraic Expression: An algebraic expression is a mathematical phrase that consists of numbers, variables, and operators (such as addition, subtraction, multiplication, and division) combined in various ways. It represents a value that can change depending on the assigned values of the variables involved. Basic examples of algebraic expressions include simple terms like $3x$ and more complex forms such as $2x + 4y - 9$.

• Components of an Algebraic Expression:

  • Variable: A letter or symbol (such as $x$ or $y$) representing an unknown numerical value that can vary. In the expression $2x + 4y - 9$, the variables are $x$ and $y$, each capable of taking on different values.

  • Constant: A fixed numerical value in an expression that does not change regardless of the values of the variables. In $2x + 4y - 9$, $-9$ serves as the constant.

  • Coefficient: The numerical factor in front of a variable, which indicates how many of the variable are being considered. Here, the coefficient of $x$ is $2$, and of $y$ is $4$, suggesting that there are two $x$'s and four $y$'s in the expression.

  • Terms: The distinct components of an expression, separated by plus (+) or minus (-) signs. The expression $2x + 4y - 9$ comprises three terms: $2x$, $4y$, and $-9$.

  • Factors of a Term: Individual numbers or variables multiplied together to form a term. Each term in an expression can be dissected into its factors; for example, the term $6xy$ consists of the factors $6$, $x$, and $y$.

  • Product: The result obtained when multiplying two or more numbers or variables together, such as the result derived from the terms that make up an expression.

Expansion of Algebraic Expressions

• Concept of Expansion: Expansion involves expressing a product involving sums or differences. It aims to rewrite expressions in a form that eliminates parentheses, employing the distributive law, which states that $a(b + c) = ab + ac$.

• Expanding Expressions with a Single Set of Brackets: Each term inside the brackets is multiplied by the term outside:

  • Example 1: Expansion of $8(x + 3)$: $\text{Expression} = 8(x + 3)$
    $\text{Result} = 8 \times x + 8 \times 3 = 8x + 24$.     

  • Example 2: $2x(5x - 7)$: $\text{Expression} = 2x(5x - 7)$ $\text{Result} = 2x \times 5x + 2x \times (-7) = 10x^2 - 14x$.     

  • Example 3: $-3(x^2 + 4x - 2)$: $\text{Expression} = -3(x^2 + 4x - 2)$ $\text{Result} = -3x^2 - 12x + 6$.

• Expanding Expressions with Two Sets of Brackets: Each term in the first bracket must be distributed to each term in the second bracket:

  • Example: $(a + 2)(a + 1)$: $\text{Expression} = (a + 2)(a + 1)$ $\text{Result} = a^2 + a + 2a + 2 = a^2 + 3a + 2$ .

• Combined Operations Including Expansion: The order of operations must adhere to the BODMAS rule:

  1. Brackets

  2. Order

  3. Division

  4. Multiplication

  5. Addition

  6. Subtraction

• Real-World Application / Problem Solving:

  • Carpet Area Problem: If Juan Maria has a carpet measuring $(3r - 2)\text{ m}$ in length and $(r + 1)\text{ m}$ in width, the area can be calculated as: $\text{Area} = \text{Length} \times \text{Width} = (3r - 2)(r + 1) = 3r^2 + 3r - 2r - 2 = (3r^2 + r - 2)\text{ m}^2$, providing a clear example of applying algebraic expansion in practical situations.

Factorization of Algebraic Expressions

• Concept of Factorization: Factorization is the inverse of expansion, where one identifies factors that can be multiplied to yield the original expression. It breaks the expression into simpler, multiplied components.

  • Factors of $3p$: Can be expressed as products like $1 \times 3p$ or $3 \times p$, yielding factors: $1$, $3$, $p$, $3p$.

• Common Factors and Highest Common Factor (HCF):

  • Common Factor: A factor that appears in two or more numbers or terms. The universal common factor is always $1$.     

  • Identifying Common Factors: For $12$ and $30$:

    • Factors of $12$ include $1, 2, 3, 4, 6, 12$.

    • Factors of $30$ include $1, 2, 3, 5, 6, 10, 15, 30$.

    • Common factors include $1, 2, 3, 6$, which are used for simplification.     

  • Finding the HCF: The highest common factor is the largest common factor found in given terms.

    • Example: For terms $6H$ and $4gsH$, the common factors are: $1, 2, H, 2H$.

    • For $9c^2d$, $3d^2$, and $6def$, the HCF is $3d$ itself.

• Methods of Factorization:

  • Method 1: Using HCF involves finding the greatest common factor to factor out from the terms.

  • Method 2: Difference of Squares applies when the expression is of the form $a^2 - b^2$, resulting in $(a + b)(a - b)$.

  • Method 3: Cross Multiplication works for quadratic expressions like $ax^2 + bx + c$.

  • Method 4: Grouping Common Factors facilitates factoring by grouping terms with shared factors together.

Example 1: Expanding a Single Bracket

Expression: $8(x + 3)$
Steps:

1. Apply the distributive property: $8(x) + 8(3)$

2. Multiply: $8x + 24$
   Result: $8x + 24$

Example 2: Expanding Two Brackets

Expression: $(a + 2)(a + 1)$
Steps:

1. Distribute each term in the first bracket to every term in the second bracket:
   
   

   • $a \times a$ gives $a^2$

   • $a \times 1$ gives $a$

   • $2 \times a$ gives $2a$

   • $2 \times 1$ gives $2$

2. Combine the results:
   $a^2 + a + 2a + 2$

3. Combine like terms: $a^2 + 3a + 2$
   Result: $a^2 + 3a + 2$

Example 3: Factoring

Expression: $x^2 + 5x + 6$
Steps:

1. Identify factors of the constant term (+6) that add up to the coefficient of the linear term (+5).
   
   

   • The factors of 6 are (1, 6) and (2, 3).

   • The pair (2, 3) adds up to 5.

2. Write the expression in factored form: $(x + 2)(x + 3)$
   Result: $(x + 2)(x + 3)$

Example 4: Solving a Quadratic Equation Using Factoring

Equation: $x^2 - 4 = 0$
Steps:

1. Factor the expression: $(x - 2)(x + 2) = 0$

2. Set each factor to zero and solve:
   
   

   • $x - 2 = 0 \rightarrow x = 2$

   • $x + 2 = 0 \rightarrow x = -2$
     Results: $x = 2$ or $x = -2$