Notes & theory for this topic

Algebra

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1. Simplify the following expressions by collecting like terms.
  1. 4uv − 2uwv + 7vuw + 5vw + wuv − 6wv
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    Group terms with the same variables (order doesn't matter, so xy = yx), then add their coefficients.
    4uv + 6uvw − vw
  2. −5xyz − 3xz − 2yxz − 4yzx − 2zxy − 3zy
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    Group terms with the same variables (order doesn't matter, so xy = yx), then add their coefficients.
    −13xyz − 3xz − 3yz
  3. 2xyz + 6xzy + 4yx − 2yzx − zxy − 2zyx
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    Group terms with the same variables (order doesn't matter, so xy = yx), then add their coefficients.
    4xy + 3xyz
  4. 6rs + 4rt + 7srt − 5str − 4tr + 3tsr
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    Group terms with the same variables (order doesn't matter, so xy = yx), then add their coefficients.
    6rs + 5rst
2. Expand the following expressions.
  1. ((((4x + 3)5x + 5)x − 3)5x + 5)
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    Expand from the innermost bracket outwards, one layer at a time.
    100x4 + 75x3 + 25x2 − 15x + 5
  2. ((((5x − 5)3x − 4)x − 4)4x + 2)
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    Expand from the innermost bracket outwards, one layer at a time.
    60x4 − 60x3 − 16x2 − 16x + 2
  3. ((((4x + 5)3x + 2)4x + 3)5x + 4)
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    Expand from the innermost bracket outwards, one layer at a time.
    240x4 + 300x3 + 40x2 + 15x + 4
  4. ((((2x − 5)3x + 4)x + 5)4x − 1)
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    Expand from the innermost bracket outwards, one layer at a time.
    24x4 − 60x3 + 16x2 + 20x − 1
3. Write the following expressions in standard form (a single polynomial with terms in descending powers).
  1. y = (x2 − 4x + 1)(x + 4)
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    Multiply every term in the second bracket by every term in the first, then collect like terms.
    y = x3 − 15x + 4
  2. y = (x2 − 4x − 6)(3x + 1)
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    Multiply every term in the second bracket by every term in the first, then collect like terms.
    y = 3x3 − 11x2 − 22x − 6
  3. y = (6x3 − 23x2 − 41x + 168) / (x − 3)
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    Factorise the numerator as (3x + 8)(2x − 7)(x − 3), cancel (x − 3), then expand.
    y = 6x2 − 5x − 56
  4. y = (2x3 − 5x2 − 19x + 42) / (2x − 7)
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    Factorise the numerator as (x − 2)(x + 3)(2x − 7), cancel (2x − 7), then expand.
    y = x2 + x − 6
4. Factorise the following expressions.
  1. y = x3 + 6x2 − 19x − 84
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    The roots are x = -3, x = -7, x = 4.
    y = (x + 3)(x + 7)(x − 4)
  2. y = x3 − 5x2 − 17x + 21
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    The roots are x = 7, x = 1, x = -3.
    y = (x − 7)(x − 1)(x + 3)
  3. y = (2x4 + 13x3 − 41x2 − 106x + 240) / (x + 3)
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    Cancel (x + 3) from the numerator. The remaining roots are x = 5/2, x = 2, x = -8.
    y = (2x − 5)(x − 2)(x + 8)
  4. y = (8x4 + 100x3 + 370x2 + 455x + 147) / (2x + 7)
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    Cancel (2x + 7) from the numerator. The remaining roots are x = -7, x = -3/2, x = -1/2.
    y = (x + 7)(2x + 3)(2x + 1)
5. Make the indicated variable the subject of each formula.
  1. Make I the subject of V = IR.
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    Divide both sides by R.
    I = V/R
  2. Make t the subject of v = u + at.
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    Subtract u from both sides, then divide by a.
    t = (v − u)/a
  3. Make rt the subject of 1/rt = 1/r1 + 1/r2 + 1/r3.
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    Add the fractions on the right over a common denominator, then take the reciprocal of both sides.
    rt = (r1r2r3) / (r2r3 + r1r3 + r1r2)
  4. Make a the subject of v2 = u2 + 2as.
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    Subtract u2, then divide by 2s.
    a = (v2 − u2)/(2s)
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