Algebraic Proof (AQA GCSE Further Maths)

Revision Note

Test Yourself
Mark

Author

Mark

Expertise

Maths

Algebraic Proof

What is algebraic proof?

  • Algebraic Proof is the process of showing something is true in every case, using algebra
  • Typical algebra skills include expanding brackets and collecting like terms
    • At the harder end, knowing the "difference of two squares" factorisation is useful

How do I prove results about odd and even numbers?

  • Assign letters (use as few letters as possible):
    • n is “any integer” (or m or k or…)
      • Integer means whole number
    • n + 1 is the consecutive integer after n (the one immediately after n)
    • 2n is an even integer (2n + 2 is the next one)
    • 2m is a different even integer (not necessarily consecutive, but any other even integer)
    • 2n + 1 is an odd integer (and 2n + 3 is the next one, or 2n - 1 is the one before, etc)
  • A "multiple of k” means it can be written as k(……), ie. k × …
  • To prove something is even, show that the algebraic result can be written as 2 × (...)
    • Make sure whatever is inside the brackets is an integer
  • To prove something is odd, show that the algebraic result can be written as 2 × (...) + 1
    • Make sure whatever is inside the brackets is an integer
  • When dealing with prime numbers, remember that primes only have factors of 1 and themselves
    • If p is prime then 1 × p or p × 1 are the only ways to write it as a product of two integers

What is the difference between an equation and an identity?

  • An equation is true for certain values only
    • For example, 3x − 1 = 5 is an equation and is only true when x = 2
    • Or another example, x2 = is an equation and is true only when = 3 or when x = −3
  • An identity is true for all values
    • For example, 2(3x) ≡ 6x is an identity because it is true for all values of x
      • Note that the symbol for an identity, , is 3 horizontal lines (like an equals sign but with an extra line)

How can I use completing the square to prove something is positive?

  • Squaring anything makes it positive...
    • ...unless its zero (which squares to zero)
  • Substituting any value of x into the expression (x - 2)2 will always give a positive output due to the "squared" outside the brackets...
    • unless the bit inside the brackets equals zero
      • x - 2 = 0, i.e. x = 2 
      • in which case, substituting in x = 2 gives zero
  • Completing the square helps to show an expression is always positive
    • f(x) = x2 - 6x + 11 can be written f(x) = (x - 3)2 + 2
      • (x - 3)≥ 0 
      • (x - 3)2 + 2 ≥  2
      • so all outputs are positive (in fact, they're greater than or equal to 2)

Exam Tip

  • It is a good idea to write a sentence at the end of your algebraic proof to say word-for-word (copied from the question) what has been proved
    • for example, "this shows that all squares of odd numbers are themselves odd"

Worked example

Prove that the difference of the squares of two consecutive even numbers is divisible by 4.

Write down an algebraic expression for an even numbe

2n 

Write down the algebraic expression for the next consecutive even number after 2

2n + 2 

Write down an expression showing the difference of the squares of two consecutive even numbers
Do the larger value subtract the smaller valu

open parentheses 2 n plus 2 close parentheses squared minus open parentheses 2 n close parentheses squared 

Expand the brackets and collect like term

open parentheses 2 n plus 2 close parentheses open parentheses 2 n plus 2 close parentheses minus 4 n squared
equals 4 n squared plus 4 n plus 4 n plus 4 minus 4 n squared
equals 8 n plus 4 

Show that the final answer is divisible by 4 (a multiple of 4)
Do this by writing it as 4 × ... and write a conclusion that copies the wording in the question

4 open parentheses 2 n plus 1 close parentheses 

is a multiple of 4, so the difference between the squares of two consecutive even numbers is divisible by 4

You've read 0 of your 0 free revision notes

Get unlimited access

to absolutely everything:

  • Downloadable PDFs
  • Unlimited Revision Notes
  • Topic Questions
  • Past Papers
  • Model Answers
  • Videos (Maths and Science)

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

Did this page help you?

Mark

Author: Mark

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.