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Flashcards · Maths · Further Algebra

Algebraic Fractions and Proof

16 cards

  1. What may you cancel in an algebraic fraction?

    Factors, never terms that are added or subtracted.

  2. What is the first step in simplifying an algebraic fraction?

    Factorise the top and the bottom.

  3. Simplify \(\dfrac{x^2 - 9}{x + 3}\) (Higher tier).

    \(x - 3\).

  4. Is \(\dfrac{x + 3}{3}\) equal to \(x\)?

    No, the 3 on top is added, so it cannot be cancelled.

  5. How do you add \(\dfrac{1}{x} + \dfrac{1}{x + 1}\) (Higher tier)?

    Use the common denominator \(x(x + 1)\) to get \(\dfrac{2x + 1}{x(x + 1)}\).

  6. How do you divide algebraic fractions (Higher tier)?

    Turn the second fraction upside down and multiply.

  7. How do you solve an equation with fractions?

    Multiply every term by the common denominator.

  8. Solve \(\dfrac{x - 1}{3} + \dfrac{x + 2}{6} = 2\).

    \(x = 4\).

  9. How do you write an even number with \(n\)?

    \(2n\).

  10. How do you write an odd number with \(n\)?

    \(2n + 1\).

  11. How do you write three consecutive numbers?

    \(n\), \(n + 1\), \(n + 2\).

  12. What does \(\equiv\) mean?

    The expressions are equal for every value of the letters, an identity.

  13. Is testing a few numbers a proof?

    No, you must use algebra to show it is true for all numbers.

  14. How do you disprove a statement?

    Find one counter-example.

  15. Give a counter-example to "\(n^2 + n + 1\) is always prime".

    \(n = 4\) gives 21, which is \(3 \times 7\).

  16. How should a proof end?

    With a sentence stating what has been shown.