Notes & theory for this topic

Algebra

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1. Simplify the following expressions by collecting like terms.
  1. 5xyz − 4xz + 6yx − yz + 2zxy − 5zy
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    Group terms with the same variables (order doesn't matter, so xy = yx), then add their coefficients.
    6xy + 7xyz − 4xz − 6yz
  2. uv + 4uwv − 7vu + 6vw − wu − 7wvu
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    Group terms with the same variables (order doesn't matter, so xy = yx), then add their coefficients.
    −6uv − 3uvw − uw + 6vw
  3. 5xyz − 5xzy + yx + 7yz + 3zx − 6zy
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    Group terms with the same variables (order doesn't matter, so xy = yx), then add their coefficients.
    xy + 3xz + yz
  4. 2rst + 5rts + 2srt − 7str − 7tr + 6ts
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    Group terms with the same variables (order doesn't matter, so xy = yx), then add their coefficients.
    2rst − 7rt + 6st
2. Expand the following expressions.
  1. ((((5x − 2)5x + 3)3x + 5)5x − 3)
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    Expand from the innermost bracket outwards, one layer at a time.
    375x4 − 150x3 + 45x2 + 25x − 3
  2. ((((3x − 2)2x − 1)3x + 3)5x − 1)
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    Expand from the innermost bracket outwards, one layer at a time.
    90x4 − 60x3 − 15x2 + 15x − 1
  3. ((((5x − 2)4x + 3)x + 4)4x − 5)
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    Expand from the innermost bracket outwards, one layer at a time.
    80x4 − 32x3 + 12x2 + 16x − 5
  4. ((((4x − 2)4x + 4)5x − 3)5x − 2)
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    Expand from the innermost bracket outwards, one layer at a time.
    400x4 − 200x3 + 100x2 − 15x − 2
3. Write the following expressions in standard form (a single polynomial with terms in descending powers).
  1. y = (3x2 + 5x + 2)(3x + 8)
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    Multiply every term in the second bracket by every term in the first, then collect like terms.
    y = 9x3 + 39x2 + 46x + 16
  2. y = (2x2 + 2x + 7)(x + 8)
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    Multiply every term in the second bracket by every term in the first, then collect like terms.
    y = 2x3 + 18x2 + 23x + 56
  3. y = (3x3 + 2x2 − 27x − 18) / (x + 3)
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    Factorise the numerator as (x − 3)(3x + 2)(x + 3), cancel (x + 3), then expand.
    y = 3x2 − 7x − 6
  4. y = (x3 + 3x2 − 24x − 80) / (x + 4)
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    Factorise the numerator as (x − 5)(x + 4)(x + 4), cancel (x + 4), then expand.
    y = x2 − x − 20
4. Factorise the following expressions.
  1. y = 6x3 + 5x2 − 44x − 15
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    The roots are x = 5/2, x = -1/3, x = -3.
    y = (2x − 5)(3x + 1)(x + 3)
  2. y = x3 + 15x2 + 68x + 96
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    The roots are x = -4, x = -3, x = -8.
    y = (x + 4)(x + 3)(x + 8)
  3. y = (4x4 + 4x3 − 23x2 + 6x + 9) / (x + 3)
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    Cancel (x + 3) from the numerator. The remaining roots are x = -1/2, x = 1, x = 3/2.
    y = (2x + 1)(x − 1)(2x − 3)
  4. y = (9x4 + 30x3 + 7x2 − 54x − 40) / (x + 1)
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    Cancel (x + 1) from the numerator. The remaining roots are x = -5/3, x = 4/3, x = -2.
    y = (3x + 5)(3x − 4)(x + 2)
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 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 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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