Notable Products

Perfect Square Trinomial

$$(a \pm b)^2 = a^2 \pm 2ab + b^2$$

Important:

$$(a - b)^2 = (b - a)^2$$

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Difference of Squares

$$a^2 - b^2 = (a + b)(a - b)$$

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Sum or Difference of Cubes

$$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$ $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

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Square of a Trinomial

$$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$$

Also:

$$(a + b - c)^2 = a^2 + b^2 + c^2 + 2ab - 2ac - 2bc$$ $$(a - b + c)^2 = a^2 + b^2 + c^2 - 2ab + 2ac - 2bc$$ $$(a - b - c)^2 = a^2 + b^2 + c^2 - 2ab - 2ac + 2bc$$ $$(a - b - c)^2 = \left( -(b + c - a) \right)^2 = (b + c - a)^2$$

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Cube of a Binomial

$$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$ $$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$

Also:

$$(a + b)^3 = a^3 + b^3 + 3ab(a + b)$$ $$(a - b)^3 = a^3 - b^3 - 3ab(a - b)$$

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Cube of a Trinomial

$$(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)$$

Another form:

$$(a + b + c)^3 = a^3 + b^3 + c^3 + 3a^2(b + c) + 3b^2(a + c) + 3c^2(a + b) + 6abc$$

Also:

$$(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a) - 3abc$$

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Product of Binomials with a Common Term

$$(x + a)(x + b) = x^2 + (a + b)x + ab$$ $$(x + a)(x + b)(x + c) = x^3 + (a + b + c)x^2 + (ab + bc + ac)x + abc$$

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Legendre’s Identities

$$(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)$$ $$(a + b)^2 - (a - b)^2 = 4ab$$ $$(a + b)^3 + (a - b)^3 = 2a(a^2 + 3b^2)$$ $$(a + b)^3 - (a - b)^3 = 2b(3a^2 + b^2)$$ $$(a + b)^4 - (a - b)^4 = 8ab(a^2 + b^2)$$

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Argan’d’s Identities

$$(x^2 + x + 1)(x^2 - x + 1) = x^4 + x^2 + 1$$ $$(x^2 + xy + y^2)(x^2 - xy + y^2) = x^4 + x^2y^2 + y^4$$

In General:

$$(x^{2m} + x^m y^n + y^{2n})(x^{2m} - x^m y^n + y^{2n}) = x^{4m} + x^{2m}y^{2n} + y^{4n}$$

Lagrange’s Identities

$$\left(a^2 + b^2\right)\left(x^2 + y^2\right) = (ax + by)^2 + (ay - bx)^2$$ $$\left(a^2 + b^2 + c^2\right)\left(x^2 + y^2 + z^2\right) = (ax + by + cz)^2 + (ay - bx)^2 + (az - cx)^2$$

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Auxiliary Identities

$$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc)$$ $$a^3 + b^3 + c^3 = \frac{1}{2}(a + b + c)\left[(a - b)^2 + (a - c)^2 + (b - c)^2\right]$$ $$a^2 + b^2 + c^2 - ab - ac - bc = \frac{1}{2}\left[(a - b)^2 + (a - c)^2 + (b - c)^2\right]$$ $$(a + b + c)^3 - a^3 - b^3 - c^3 = 3(a + b)(b + c)(c + a)$$ $$(a + b)(b + c)(a + c) + abc = (a + b + c)(ab + ac + bc)$$ $$(a - b)^2 + (b - c)^2 + (a - c)^2 = (a^2 + b^2 + c^2) - 2(ab + ac + bc)$$

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Conditional Equalities

• If $a + b + c = 0$, then:

$$a^2 + b^2 + c^2 = -2(ab + ac + bc)$$ $$a^3 + b^3 + c^3 = 3abc$$ $$a^4 + b^4 + c^4 = 2\left(a^2b^2 + a^2c^2 + b^2c^2\right)$$ $$a^5 + b^5 + c^5 = -5abc(ab + ac + bc)$$ $$(ab + ac + bc)^2 = a^2b^2 + a^2c^2 + b^2c^2$$

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Notable Implications

• If $\frac{a}{b} + \frac{b}{a} = 2$, then:

$$a = b$$

• If $a^2 + b^2 + c^2 = ab + ac + bc$, then:

$$a = b = c$$

• If $a^3 + b^3 + c^3 = 3abc$ and $a + b + c = 0$, then:

$$a = b = c = 0$$

• If $a^2 + b^2 + c^2 + \dots + z^2 = 0$, then:

$$a = b = c = \dots = z = 0$$

• If $\sqrt[n]{a} + \sqrt[n]{b} + \dots + \sqrt[n]{z} = 0$, then:

$$a = b = \dots = z = 0$$

• If $x + x^{-1} = a$, then:

$$x^2 + x^{-2} = a^2 - 2$$ $$x^3 + x^{-3} = a^3 - 3a$$ $$x^4 + x^{-4} = (a^2 - 2)^2 - 2$$