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Algebraic Techniques & Manipulation Flashcards

7 cards from real TMUA practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

Read the first 7 Algebraic Techniques & Manipulation flashcards as text

  1. If x - 1/x = 3, find x² + 1/x².

    Answer: 11

    Squaring x - 1/x = 3 gives x² - 2 + 1/x² = 9, so x² + 1/x² = 11.

  2. Find the remainder when x³ - 3x² + 5x - 2 is divided by (x - 2).

    Answer: 4

    By the remainder theorem, substitute x=2: 8 - 12 + 10 - 2 = 4.

  3. Solve for real x: x/(x-2) + 2/(x+1) = 1

    Answer: x = 0 or x = 3

    Multiplying through by (x-2)(x+1) and simplifying yields x²-3x=0, so x=0 or x=3.

  4. Simplify: (√5 + √3)² - (√5 - √3)²

    Answer: 4√15

    Using a²-b² = (a+b)(a-b): [(√5+√3)+(√5-√3)][(√5+√3)-(√5-√3)] = (2√5)(2√3) = 4√15.

  5. For what values of c does x² + 6x + c > 0 for all real x?

    Answer: c > 9

    The quadratic is always positive when discriminant 9.

  6. Factorise: x³ + 8

    Answer: (x+2)(x²-2x+4)

    This is a sum of cubes: a³+b³ = (a+b)(a²-ab+b²) with a=x, b=2.

  7. If f(x) = (x+1)/(x-1), find f(f(x)).

    Answer: x

    f(f(x)) = [(x+1)/(x-1)+1]/[(x+1)/(x-1)-1] = [2x/( x-1)]/[2/(x-1)] = x.