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1.1~1.12 - Coggle Diagram

- - - - Three ways that letters could represent numbers
        
        1.
        
        Letters could represent any type of number
        
        a+b = b+a
        
        2.
        
        Letters could represent a specific type of number
        
        S = πr * πr
        
        3.
        
        Letters could represent a variable in an equation
        
        35 - 2(2X + 3) = 7
      - Tips & Writing Rules
        
        When a number times a letter, the multiplication sign can be
        removed. But the number must be put in front.
        
        If the number is a mixed fraction, it must be turned into an improper fraction.
        
        Usually a division is written as a fraction.
  - - - Given that the value of 2x − 5x − 5 is 3. Find the value of 4x − 10x.
        ∵ 2x − 5x − 5 = 3 ∴ 2x − 5x = 8 ∴ 4x − 10x = 16
      - Substitute numerical values for letters in algebraic expressions according to the
        result calculated from the operational relationship is called the value of the algebraic expression.
  - - - What is an algebraic expression?
        
        An algebraic expression is an expression built up from constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number).
        
        A single number or letter is also an algebraic expression.
      - Be Careful!
        
        An equation or inequality is NOT an algebraic expression.
- - - - What is like terms?
        
        Like terms are terms with exactly the same variable form.
      - Combining like terms
        
        The result of adding the coefficients of similar items is used as the coefficient after merging similar items, and the letters and their exponents remain unchanged.
      - Ex. 8x(2) + 2x(2) = 10x(2)
  - - - What is a Polynomial?
        
        A polynomial is the sum of monomials.
      - What is a Polynomial?(At a whole)
        
        A polynomial(at a whole) is both polynomials and monomials.
      - What is a Monomial?
        
        A monomial is a product of a number and one or more variables with non-negative integer exponents.
        
        ESP. A number or a variable is also a monomial.
      - 1.4 Key words
        
        Term
        
        In Algebra a term is either a single number or variable, or numbers and variables multiplied together.
        
        coefficient
        
        The coefficient of a monomial is the constant term of the monomial.
        
        Degree
        
        Degree of a polynomial
        
        The degree of a polynomial is the highest degree on one of its terms
        
        Ex. 3y(2)x + z(5)x(3) + 3's degree is 5 + 3 = 8
        
        Degree of a monomial
        
        The degree of a monomial is the sum of all its degrees on all its terms.
        
        Ex. 3y(2)x(5)'s degree is 2 + 5 = 7
        
        Constant
        
        A single number is called a constant term, or a constant. It is a type of term.
      - Tips & Writing Rules:
        
        Remember: A fraction that has a letter in its Denominator is not a monomial/polynomial/polynomial(at a whole)
        
        Write Polynomials in Ascending and Descending Order: The terms of a polynomial are usually arranged so that the powers of one variable are in descending (decreasing) order or ascending (increasing) order
      - Names of Degrees & Terms
        
        Linear
        
        Names of Degrees
        
        Quadratic
        
        Cubic
        
        Quartic
        
        Quintic
        
        Monomial
        
        Name of Terms
        
        Binomial
        
        Trinomial
        
        Ex. A cubic binomial - x(2)y
  - - - How to add/subtract?
        
        The addition and subtraction of integers is the addition and subtraction of monomials and polynomials. The addition and subtraction of integers can be completed by removing brackets and combining like terms.
      - The rules to remove brackets
        
        If a bracket is preceded by a plus sign, remove it by
        writing its terms as they are.
        
        If a bracket is preceded by a minus sign, change
        positive signs within it to negative and vice-versa.
        
        Ex. x + (y + z) = x + y + z
        
        Ex. x – (y + z) = x – y – z
- - - - Formula
        
        (ab)(n) = a(n)b(n)
  - - - Formula
        
        For any number a and all integers m and n, a(m) • a(n) = a(m+n).
        To multiply two powers that have the same base, add their
        exponents.
      - Tips
        
        If n is odd, then (-a)(n) [meant a power here of n] = -a(n)
        
        If n is even, then (-a)(n) [meant a power here of n] = a(n)
      - Steps to calculate:
        
        Determine the sign (whether positive or negative) and unify the base
        
        Calculate
        
        If the bases are opposites of each other, look at the exponent.
  - - - Formula
        
        [ a(m) ] x (n) = a(mn) = [ a(n) ] x (m)
        
        When raising a power to a power, keep the
        base and multiply the indices.
      - Short Summary
        
        Operation type
        
        Power of a power
        
        formula
        
        operations in rules
        
        Power
        
        Calculation results(bases, power)
        
        Same base, power * power.
- - - - (a+b)(a-b) = a(2) + b(2)
      - Characteristics
        
        The product of sum and difference of the same two terms is the difference of their squares.
  - - - Multiplying monomials
        
        Steps to calculate
        
        Step1: Multiply the coefficients.
        
        Step2: Multiply powers with same base.
        
        Step3: Simplify.
        
        Tips
        
        Be careful with the signs - you should always look at the sign in front of each term.
        
        Order matters! Always think of BODMAS (Brackets --> Order --> Division&Multiplication --> Addition&Subtraction)
        
        Summary of steps:
        
        Step0: Power of product/power.
        
        Step1: Multiply the coefficients.
        
        Step2: Multiply powers with same base.
        
        Step3: Simplify
    - - Multiplying a monomial by a polynomial
        
        General steps:
        
        Step1: Multiply all terms of the polynomial by the monomial. (Multiply coefficients and powers with the same base)
        
        Step2: Add all the products.
        
        Order MATTERS!
    - - Multiplying polynomials
        
        Formula
        
        (a+m)(b+n) = a(b+n) + m(b+n) = ab+an+bm+mn
        
        Repeated use of distributive law of multiplication
        
        S1. Multiply Polynomials
        
        S2. Multiply a Polynomial by a Monomial
        
        S3. Multiply Monomials
        
        Steps
        
        Step1: Multiply all terms of one polynomial by each term of the other polynomial.
        
        Step2: Add all the products.