Chapter 7 - Algebraic Methods

• What is the first step in simplifying algebraic fractions?


  Factorise the numerator and denominator where possible
• How do you cancel common factors in algebraic fractions?


  By dividing both the numerator and denominator by the common factors
• What is an example of a polynomial?


  4x³ + 3x - 9
• What is the definition of a polynomial?


  A polynomial is a finite expression with positive whole number indices
• How can you divide a polynomial by (x ± p)?


  By using long division
• What does the factor theorem state?


  If f(p) = 0, then (x-p) is a factor of f(x)
• What is the relationship between f(p) and factors of f(x) according to the factor theorem?


  If (x-p) is a factor of f(x), then f(p) = 0
• Why is it important to differentiate between the two statements of the factor theorem?


  Because they imply different conditions for factors and roots
• What are the steps to use the factor theorem to factorise a cubic function g(x)?


  1. Substitute values into g(x) until g(p) = 0.
  2. Divide g(x) by (x-p).
  3. Write g(x) = (x-p)(ax² + bx + c).
  4. Factorise the quadratic factor if possible.
• What is a mathematical proof?


  A logical and structured argument to show that a mathematical statement is always true
• What is the final step in a mathematical proof?


  A statement of what has been proven
• What is a theorem?


  A statement that has been proven
• What is a conjecture?


  A statement that has yet to be proven
• How can you prove a mathematical statement is true by deduction?


  By starting from known facts or definitions and using logical steps
• What is an example of proof by deduction?


  The product of two odd numbers is odd
• Why is showing one case not sufficient for a proof?


  Because one example does not prove the statement for all cases
• What are the key points to remember when constructing a mathematical proof?


  • State any information or assumptions used
  • Show every step clearly
  • Ensure logical progression of steps
  • Cover all possible cases
  • Write a statement of proof at the end
• What is the identity for the difference of squares?


  (a+b)(a−b) ≡ a² − b²
• How do you prove an identity?


  By starting with one side and manipulating it to match the other side
• What does the symbol ≡ mean?


  It means 'is always equal to'
• What should you avoid when proving an identity?


  Don't try to 'solve' an identity like an equation
• How can you prove a mathematical statement by exhaustion?


  By breaking the statement into smaller cases and proving each case separately
• What is a counter-example?


  One example that does not work for the statement
• How many examples do you need to disprove a statement?


  One example is sufficient to disprove a statement
• What are the main methods of proof in mathematics?

  1. Proof by deduction
  2. Proof by exhaustion
  3. Proof by counter-example