Algebraic proof

• What is algebraic proof?

  Algebraic proof is the process of showing something is true in every case, using algebra.
• What are typical algebra skills involved in algebraic proof?

  Typical algebra skills include expanding brackets and collecting like terms.
• Why is knowing the "difference of two squares" factorization useful in algebraic proof?

  It is useful because it helps in simplifying expressions and proving identities.
• What does the letter 'n' represent in algebraic proofs about odd and even numbers?

  'n' represents any integer.
• What is the definition of an integer?

  An integer means a whole number.
• What is the expression for the consecutive integer after 'n'?

  The consecutive integer after 'n' is 'n + 1'.
• How can you express an even integer in terms of 'n'?

  An even integer can be expressed as '2n'.

  
• What is the expression for the next even integer after '2n'?

  The next even integer after '2n' is '2n + 2'.

  
• How can you express an odd integer in terms of 'n'?

  An odd integer can be expressed as '2n + 1'.

  
• What does "a multiple of k" mean?

  A multiple of k means it can be written as k(…), i.e., k × …
• How do you prove that an algebraic result is even?

  To prove something is even, show that the algebraic result can be written as '2 × (...)'.
• How do you prove that an algebraic result is odd?

  To prove something is odd, show that the algebraic result can be written as '2 × (...) + 1'.
• What is a key characteristic of prime numbers?

  Prime numbers only have factors of 1 and themselves.
• How can a prime number 'p' be expressed as a product of two integers?

  If 'p' is prime, then it can be expressed as '1 × p' or 'p × 1'.
• What is the difference between an equation and an identity?

  An equation is true for certain values only, while an identity is true for all values.
• Give an example of an equation.

  3x − 1 = 5 is an example of an equation.

  
• What is an example of an identity?

  2(3x) ≡ 6x is an example of an identity.
• What symbol is used to denote an identity?

  The symbol for an identity is three horizontal lines (≡).
• What is a good practice to include at the end of an algebraic proof?


  • Write a sentence that copies the wording from the question.
  • For example, "this shows that all squares of odd numbers are themselves odd."

  
• How do you prove that the difference of the squares of two consecutive even numbers is divisible by 4?

  Write down the algebraic expression for an even number, then for the next consecutive even number, and show the difference of their squares is a multiple of 4.
• What is the algebraic expression for an even number used in the proof of the difference of squares?

  The algebraic expression for an even number is '2n'.

  
• What is the algebraic expression for the next consecutive even number after '2n'?

  The algebraic expression for the next consecutive even number after '2n' is '2n + 2'.

  
• How do you show that the difference of the squares of two consecutive even numbers is divisible by 4?

  Expand the brackets, collect like terms, and show the final answer is divisible by 4.
• What should you do at the conclusion of your proof regarding the difference of squares?

  Write a conclusion that copies the wording in the question, stating it is a multiple of 4.