Linear Equations and Polynomial Expressions

Linear Equations

• Definition: An equation of the form ( ax + b = 0 ) where ( a ) and ( b ) are constants, and ( x ) is the variable.
• Standard Form: Typically written as ( y = mx + c ) where:
  • ( m ): slope of the line (change in ( y ) per unit change in ( x ))
  • ( c ): y-intercept (value of ( y ) when ( x = 0 ))
• Types:
  • One-variable: ( ax + b = 0 )
  • Two-variable: ( ax + by + c = 0 )
• Graph: A straight line; the slope ( m ) determines the inclination.
• Solving Techniques:
  • Isolate the variable (e.g., ( x )).
  • Use inverse operations (addition/subtraction, multiplication/division).
  • Graphical methods: Plotting points to find the intersection with the line.

Polynomial Expressions

• Definition: An expression involving sums and/or products of variables raised to non-negative integer powers.
• Standard Form: Written as ( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 ), where:
  • ( a_n, a_{n-1}, ..., a_0 ): coefficients (real numbers)
  • ( n ): degree of the polynomial (highest power of ( x ))
• Types:
  • Monomial: One term (e.g., ( 3x^2 ))
  • Binomial: Two terms (e.g., ( x^2 + 4x ))
  • Trinomial: Three terms (e.g., ( x^2 + 2x + 1 ))
• Operations:
  • Addition/Subtraction: Combine like terms.
  • Multiplication: Use distributive property or the FOIL method for binomials.
  • Division: Polynomial long division or synthetic division.
• Factoring:
  • Common methods: Factoring out the greatest common factor, grouping, difference of squares, and trinomial factoring.

Factoring Expressions

• Factoring involves rewriting expressions as a product of simpler expressions.
• Greatest Common Factor (GCF): Identify the highest common factor among terms and factor it out.
• Quadratic Trinomials: Factor using methods like grouping or applying the formula (ax2+bx+c=(px+q)(rx+s))(ax^2 + bx + c = (px + q)(rx + s))(ax2+bx+c=(px+q)(rx+s)).
• Example: Factoring 6x² + 9x yields 3x(2x + 3).

Simplifying Fractions

• Simplifying fractions involves reducing them to their lowest terms.
• Find the greatest common factor (GCF) of the numerator and denominator.
• Divide both numerator and denominator by the GCF.
• Example: Simplifying 8/12, the GCF is 4. Dividing both by 4 results in 2/3.

Distributive Property

• The distributive property states that a(b + c) = ab + ac.
• Key for expanding expressions and simplifying them.
• Multiply the term outside parentheses by each term inside.
• Example: 2(x + 3) expands to 2x + 6.

Using Exponents

• Product of Powers: Multiplying powers with the same base involves adding exponents: (am⋅an=am+n)(a^m \cdot a^n = a^{m+n})(am⋅an=am+n)
• Quotient of Powers: Dividing powers with the same base involves subtracting exponents: (am/an=am−n)( a^m / a^n = a^{m-n} )(am/an=am−n) (where a ≠ 0).
• Power of a Power: Raising a power to another power involves multiplying exponents: ((am)n=am⋅n)( (a^m)^n = a^{m \cdot n} )((am)n=am⋅n)
• Zero Exponent: Any non-zero number raised to the power of zero equals 1: (a0=1)( a^0 = 1 )(a0=1) (where a ≠ 0).
• Simplify expressions by applying these laws: (x2⋅x3=x2+3=x5)(x^2 \cdot x^3 = x^{2+3} = x^5)(x2⋅x3=x2+3=x5)

Evaluating Expressions

• Replace variables with their corresponding numerical values (e.g., for (3x + 2) if (x = 2): (3(2) + 2 = 8)).
• Follow the order of operations (PEMDAS/BODMAS) to ensure accurate results (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).
• Differentiate between purely numerical expressions and algebraic expressions with variables.

Understanding Variables

• Variables represent unknown quantities, often denoted by letters like (x), (y), and (z).
• Variables can take on different values depending on the context, making them flexible tools in mathematics.
• Variables are used to create algebraic expressions that model real-world situations and relationships.

Order of Operations

• PEMDAS/BODMAS dictates the sequence of operations when evaluating expressions.
• Parentheses take priority, signifying the order of operations within the expression.
• Following the order of operations ensures consistent and accurate results in simplifying and evaluating expressions.

Combining Like Terms

• Like terms share the same variable raised to the same power, for example, (4a) and (-2a).
• Combining like terms involves grouping them together and then adding or subtracting their coefficients.
• For example, in the expression (4a + 3b - 2a + b), the like terms (4a) and (-2a) can be combined to get (2a), and (3b) and (b) can be combined to get (4b), resulting in (2a + 4b).

Order Of Operations

• To correctly evaluate expressions, follow the hierarchy of operations, commonly remembered by the acronym PEMDAS or BODMAS.
• PEMDAS/BODMAS stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
• For example, in (3 + 6 \times (5 + 4) ÷ 3 - 7), simplifying the expression within the parentheses first gives (3 + 6 \times 9 ÷ 3 - 7).
• Following the order of operations leads to (3 + 54 ÷ 3 - 7), then (3 + 18 - 7), and finally (14).

Evaluating Expressions

• Evaluating an expression involves substituting specific values for variables and calculating the result.
• This process involves substituting each variable with its corresponding value and then following the order of operations.
• For example, to evaluate (2x + 3y) when (x = 2) and (y = 3), first substitute to get (2(2) + 3(3)), then simplify according to PEMDAS to get (4 + 9), which equals (13).

Quadratic Equations: Factoring Techniques

• A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Factoring involves breaking down a quadratic expression into simpler expressions (factors) that, when multiplied, result in the original expression.

Factoring Quadratic Equations

• Step 1: Ensure the quadratic equation is in standard form (ax² + bx + c = 0).
• Step 2: Identify if a, b, or c share a greatest common factor (GCF) and factor it out if applicable.
• Step 3: Calculate the product of a and c (ac) and find two numbers that multiply to ac and add up to b (the coefficient of x).
• Step 4: Use the two numbers found to split the middle term (bx) into two terms.
• Step 5: Group the terms into two pairs and factor out the common factor from each pair. Combine the common binomial factor.
• Step 6: The factored form will be in the format (px + q)(rx + s) = 0, where p, q, r, and s are constants.
• Step 7: Set each factor equal to zero and solve for x to find the possible solutions.

Special Cases

• Perfect Square Trinomial: Recognize patterns like (x + a)² = x² + 2ax + a².
• Difference of Squares: Use the identity a² - b² = (a - b)(a + b) to solve equations in this form.

Tips for Factoring

• Practice various forms of quadratics for greater familiarity.
• Check solutions by substituting them back into the original equation.