Algebraic Identities — CDS Elementary Mathematics

Formulas (3)

• Everything from a + b and ab · Cube of a sum
  $$a^3 + b^3 = (a + b)^3 - 3ab(a + b)$$
• Spotting an expanded cube · Cube of a difference
  $$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$
• Difference of squares, sum and difference of cubes · Sum and difference of cubes
  $$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$$

Watch out for (3)

• The correction term is 3ab(a + b)
  → Everything from a + b and ab
• Check the middle coefficients
  → Spotting an expanded cube
• The signs pair up opposite
  → Difference of squares, sum and difference of cubes

Formulas (2)

• Squaring up and down the ladder · Squaring a reciprocal sum
  $$\left(x \pm \tfrac1x\right)^2 = x^2 + \tfrac{1}{x^2} \pm 2$$
• Cubes and higher powers · Cubes of a reciprocal sum
  $$x^3 + \tfrac{1}{x^3} = u^3 - 3u, \qquad x^3 - \tfrac{1}{x^3} = v^3 + 3v$$

Watch out for (2)

• x − 1/x has two signs
  → Squaring up and down the ladder
• Plus 3v for the difference, minus 3u for the sum
  → Cubes and higher powers

Formulas (3)

• The square of a + b + c · Square of a sum of three
  $$(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$$
• Expanding a product of brackets · Cubic from its roots
  $$(x - a)(x - b)(x - c) = x^3 - \textstyle\sum a\, x^2 + \sum ab\, x - abc$$
• When 2s = a + b + c · Semi-perimeter pieces
  $$(s - a) + (s - b) + (s - c) = s$$

Watch out for (3)

• A square root gives two signs
  → The square of a + b + c
• The signs alternate
  → Expanding a product of brackets
• s is half the perimeter
  → When 2s = a + b + c

Formulas (2)

• When three numbers add to zero · Zero sum
  $$a + b + c = 0 \;\Rightarrow\; a^3 + b^3 + c^3 = 3abc$$
• The full factorisation · Cube identity
  $$a^3 + b^3 + c^3 - 3abc = (a + b + c)\left[(a + b + c)^2 - 3(ab + bc + ca)\right]$$

Watch out for (2)

• Check that the three really add to zero
  → When three numbers add to zero
• Zero does not mean a + b + c = 0
  → The full factorisation

Formulas (3)

• Sums of squares that vanish · Sum of squared differences
  $$a^2 + b^2 + c^2 - ab - bc - ca = \tfrac12\left[(a - b)^2 + (b - c)^2 + (c - a)^2\right]$$
• Least values: t + 1/t is at least 2 · AM–GM
  $$pt + \dfrac{q}{t} \ge 2\sqrt{pq} \quad (p, q, t > 0)$$
• Which is larger? · Mean of squares against square of mean
  $$\dfrac{a^2 + b^2}{2} - \left(\dfrac{a + b}{2}\right)^2 = \dfrac{(a - b)^2}{4} \ge 0$$

Watch out for (3)

• Read what the stem already gives
  → Sums of squares that vanish
• Only for positive values
  → Least values: t + 1/t is at least 2
• Greater or greater-or-equal?
  → Which is larger?

Formulas (3)

• Rewrite each piece with the condition · A typical rewrite
  $$ab + bc + ca = 0 \;\Rightarrow\; a^2 - bc = a(a + b + c)$$
• Equal ratios · Adding equal ratios
  $$\dfrac ab = \dfrac cd = \dfrac ef = k \;\Rightarrow\; k = \dfrac{a + c + e}{b + d + f}$$
• Factor first, then substitute · Divide out the known factor
  $$x^2 + xy - 2y^2 = (x + 2y)(x - y)$$

Watch out for (3)

• Test with numbers that fit
  → Rewrite each piece with the condition
• Do not drop the zero-sum case
  → Equal ratios
• Dividing by something that could be zero
  → Factor first, then substitute

Formulas (3)

• Factorise and cancel · Cancel a common factor
  $$\dfrac{(x - 1)(x^2 + x + 1)}{x^2 + x + 1} = x - 1$$
• Combining fractions · Two fractions
  $$\dfrac{1}{x - 1} - \dfrac{1}{x + 1} = \dfrac{2}{x^2 - 1}$$
• Equations with fractions · Excluded values
  $$\dfrac{P(x)}{Q(x)} = 0 \;\Rightarrow\; P(x) = 0,\ Q(x) \ne 0$$

Watch out for (3)

• Cancel factors, not terms
  → Factorise and cancel
• Order of subtraction
  → Combining fractions
• Check the denominators
  → Equations with fractions

Formulas (2)

• Three cyclic fraction sums · The key cyclic sum
  $$\dfrac{a^2}{(a - b)(a - c)} + \dfrac{b^2}{(b - c)(b - a)} + \dfrac{c^2}{(c - a)(c - b)} = 1$$
• Factors of cyclic expressions · The basic cyclic factorisation
  $$a(b^2 - c^2) + b(c^2 - a^2) + c(a^2 - b^2) = (a - b)(b - c)(c - a)$$

Watch out for (2)

• Watch the order of the factors
  → Three cyclic fraction sums
• One test disproves; it does not prove
  → Factors of cyclic expressions