Notes & theory for this topic

Algebra

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1. Simplify the following expressions by collecting like terms.
  1. 5xyz − xz + 5yxz − 3yz + 8zx + 4zy
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    Group terms with the same variables (order doesn't matter, so xy = yx), then add their coefficients.
    10xyz + 7xz + yz
  2. 7uvw + 4uwv + 3vuw + 8vwu + 6wu + 3wvu
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    Group terms with the same variables (order doesn't matter, so xy = yx), then add their coefficients.
    25uvw + 6uw
  3. −xyz − 5xzy + yx + 2yz − 6zxy + 3zyx
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    Group terms with the same variables (order doesn't matter, so xy = yx), then add their coefficients.
    xy − 9xyz + 2yz
  4. rst + 3rts − 5srt + 2st + 3trs − 3tsr
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    Group terms with the same variables (order doesn't matter, so xy = yx), then add their coefficients.
    −rst + 2st
2. Expand the following expressions.
  1. ((((3x − 5)5x − 3)5x − 1)2x − 4)
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    Expand from the innermost bracket outwards, one layer at a time.
    150x4 − 250x3 − 30x2 − 2x − 4
  2. ((((5x + 2)5x + 5)2x + 5)4x + 2)
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    Expand from the innermost bracket outwards, one layer at a time.
    200x4 + 80x3 + 40x2 + 20x + 2
  3. ((((3x − 4)3x − 5)5x − 5)5x + 5)
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    Expand from the innermost bracket outwards, one layer at a time.
    225x4 − 300x3 − 125x2 − 25x + 5
  4. ((((2x − 5)x + 2)2x + 4)4x − 2)
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    Expand from the innermost bracket outwards, one layer at a time.
    16x4 − 40x3 + 16x2 + 16x − 2
3. Write the following expressions in standard form (a single polynomial with terms in descending powers).
  1. y = (3x2 − 2x + 6)(3x + 3)
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    Multiply every term in the second bracket by every term in the first, then collect like terms.
    y = 9x3 + 3x2 + 12x + 18
  2. y = (x2 − 2x − 5)(2x − 7)
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    Multiply every term in the second bracket by every term in the first, then collect like terms.
    y = 2x3 − 11x2 + 4x + 35
  3. y = (4x3 + 16x2 − 79x + 35) / (x + 7)
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    Factorise the numerator as (2x − 1)(2x − 5)(x + 7), cancel (x + 7), then expand.
    y = 4x2 − 12x + 5
  4. y = (6x3 + 29x2 + 30x + 7) / (2x + 7)
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    Factorise the numerator as (3x + 1)(x + 1)(2x + 7), cancel (2x + 7), then expand.
    y = 3x2 + 4x + 1
4. Factorise the following expressions.
  1. y = 2x3 + 9x2 − 77x − 294
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    The roots are x = 6, x = -7, x = -7/2.
    y = (x − 6)(x + 7)(2x + 7)
  2. y = x3 + 15x2 + 48x − 64
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    The roots are x = -8, x = -8, x = 1.
    y = (x + 8)(x + 8)(x − 1)
  3. y = (x4 + 13x3 + 28x2 − 132x − 288) / (x + 8)
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    Cancel (x + 8) from the numerator. The remaining roots are x = 3, x = -2, x = -6.
    y = (x − 3)(x + 2)(x + 6)
  4. y = (3x4 − x3 − 9x2 − 3x + 2) / (x + 1)
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    Cancel (x + 1) from the numerator. The remaining roots are x = 2, x = -1, x = 1/3.
    y = (x − 2)(x + 1)(3x − 1)
5. Make the indicated variable the subject of each formula.
  1. Make R the subject of V = IR.
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    Divide both sides by I.
    R = V/I
  2. Make a the subject of v = u + at.
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    Subtract u from both sides, then divide by t.
    a = (v − u)/t
  3. Make ct the subject of 1/ct = 1/c1 + 1/c2 + 1/c3.
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    Add the fractions on the right over a common denominator, then take the reciprocal of both sides.
    ct = (c1c2c3) / (c2c3 + c1c3 + c1c2)
  4. Make s the subject of v2 = u2 + 2as.
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    Subtract u2, then divide by 2a.
    s = (v2 − u2)/(2a)
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