Special Products

Perfect Square Trinomial

$$\boxed{(a + b)^2 = a^2 + 2ab + b^2}$$

$$\boxed{(a - b)^2 = a^2 - 2ab + b^2}$$

Examples

• $(x + 3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9$
• $(2a - 5)^2 = (2a)^2 - 2(2a)(5) + 5^2 = 4a^2 - 20a + 25$
• $(y + 1)^2 = y^2 + 2(y)(1) + 1^2 = y^2 + 2y + 1$
• $(3x - 2y)^2 = (3x)^2 - 2(3x)(2y) + (2y)^2 = 9x^2 - 12xy + 4y^2$
• $(a + \frac{1}{2})^2 = a^2 + 2(a)(\frac{1}{2}) + (\frac{1}{2})^2 = a^2 + a + \frac{1}{4}$
• $(5 - b)^2 = 5^2 - 2(5)(b) + b^2 = 25 - 10b + b^2$

Difference of Squares

$$\boxed{a^2 - b^2 = (a + b)(a - b)}$$

Examples

• $x^2 - 9 = x^2 - 3^2 = (x + 3)(x - 3)$
• $4a^2 - 25 = (2a)^2 - 5^2 = (2a + 5)(2a - 5)$
• $16y^2 - 1 = (4y)^2 - 1^2 = (4y + 1)(4y - 1)$
• $49 - z^2 = 7^2 - z^2 = (7 + z)(7 - z)$
• $\frac{x^2}{4} - \frac{y^2}{9} = (\frac{x}{2})^2 - (\frac{y}{3})^2 = (\frac{x}{2} + \frac{y}{3})(\frac{x}{2} - \frac{y}{3})$
• $(x + 2)^2 - 16 = (x + 2)^2 - 4^2 = (x + 2 + 4)(x + 2 - 4) = (x + 6)(x - 2)$

Square of a Trinomial

$$\boxed{(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc}$$

$$\boxed{(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc)}$$

Examples

• $(x + y + 2)^2 = x^2 + y^2 + 4 + 2xy + 4x + 4y$
• $(a + 3 + b)^2 = a^2 + 9 + b^2 + 6a + 2ab + 6b$
• $(1 + m + n)^2 = 1 + m^2 + n^2 + 2m + 2n + 2mn$
• $(2a + 3b + 1)^2 = 4a^2 + 9b^2 + 1 + 12ab + 4a + 6b$
• $(x - 1 + y)^2 = x^2 + 1 + y^2 - 2x + 2xy - 2y$
• $(p + q + r)^2 = p^2 + q^2 + r^2 + 2pq + 2pr + 2qr$

Cube of a Binomial

$$\boxed{(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3}$$

$$\boxed{(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3}$$

Examples

• $(x + 2)^3 = x^3 + 3x^2(2) + 3x(4) + 8 = x^3 + 6x^2 + 12x + 8$
• $(a - 1)^3 = a^3 - 3a^2(1) + 3a(1) - 1 = a^3 - 3a^2 + 3a - 1$
• $(2x + y)^3 = (2x)^3 + 3(4x^2)(y) + 3(2x)(y^2) + y^3 = 8x^3 + 12x^2y + 6xy^2 + y^3$
• $(a - 3b)^3 = a^3 - 3a^2(3b) + 3a(9b^2) - 27b^3 = a^3 - 9a^2b + 27ab^2 - 27b^3$
• $(1 + x)^3 = 1 + 3x + 3x^2 + x^3$
• $(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$

Product of Binomials with a Common Term (Stevin’s Identity)

$$\boxed{(x + a)(x + b) = x^2 + (a + b)x + ab}$$

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

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

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

Examples

• $(x + 3)(x + 5) = x^2 + (3 + 5)x + 3 \cdot 5 = x^2 + 8x + 15$
• $(x + 2)(x - 7) = x^2 + (2 - 7)x + 2 \cdot (-7) = x^2 - 5x - 14$
• $(x - 4)(x - 6) = x^2 + (-4 - 6)x + (-4) \cdot (-6) = x^2 - 10x + 24$
• $(x + 1)(x + 9) = x^2 + (1 + 9)x + 1 \cdot 9 = x^2 + 10x + 9$
• $(x - 3)(x + 8) = x^2 + (-3 + 8)x + (-3) \cdot 8 = x^2 + 5x - 24$
• $(x + a)(x + b) = x^2 + (a + b)x + ab$

Cube of a Trinomial

$$\boxed{(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)}$$

Examples

• $(x + 1 + 2)^3 = x^3 + 1^3 + 2^3 + 3(x + 1)(1 + 2)(2 + x) = x^3 + 9 + 9(x + 1)(x + 2)$
• $(1 + 2 + 3)^3 = 1^3 + 2^3 + 3^3 + 3(1 + 2)(2 + 3)(3 + 1) = 36 + 180 = 216$
• $(a + 0 + b)^3 = a^3 + b^3 + 3ab(a + b) = a^3 + 3a^2b + 3ab^2 + b^3$
• $(x + y + 0)^3 = x^3 + y^3 + 3xy(x + y)$
• $(1 + x + 1)^3 = (x + 2)^3 = x^3 + 8 + 6(x + 1)^2$
• $(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)$

Sum and Difference of Cubes

$$\boxed{a^3 + b^3 = (a + b)(a^2 - ab + b^2)}$$

$$\boxed{a^3 - b^3 = (a - b)(a^2 + ab + b^2)}$$

Examples

• $8 + 27 = 2^3 + 3^3 = (2 + 3)(4 - 6 + 9) = 5 \cdot 7 = 35$
• $x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)$
• $64 + 125 = 4^3 + 5^3 = (4 + 5)(16 - 20 + 25) = 9 \cdot 21 = 189$
• $27a^3 - 64b^3 = (3a)^3 - (4b)^3 = (3a - 4b)(9a^2 + 12ab + 16b^2)$
• $1 + x^3 = (1 + x)(1 - x + x^2)$
• $y^3 + 1000 = y^3 + 10^3 = (y + 10)(y^2 - 10y + 100)$

Argand’s Identities

$$\boxed{(a^2 + a + 1)(a^2 - a + 1) = a^4 + a^2 + 1}$$

$$\boxed{(a^2 + ab + b^2)(a^2 - ab + b^2) = a^4 + a^2b^2 + b^4}$$

Gauss’s Identity

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

$$(a + b)(b + c)(a + c) + abc = (a + b + c)(ab + ac + bc)$$

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

Lagrange’s Identities

$$(ax + by)^2 + (ay - bx)^2 = (a^2 + b^2)(x^2 + y^2)$$

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

Explicit Forms of $a^n + b^n$

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

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

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

$$a^5 + b^5 = (a + b)^5 - 5ab(a + b)^3 + 5(ab)^2(a + b)$$

Conditional Identities

If $a + b + c = 0$, then:

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

$$a^3 + b^3 + c^3 = 3abc$$

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

$$(a^2 + b^2 + c^2)^2 = 2(a^4 + b^4 + c^4)$$

$$a^4 + b^4 + c^4 = 2(a^2b^2 + a^2c^2 + b^2c^2) = \frac{1}{2}(a^2 + b^2 + c^2)^2$$

$$a^5 + b^5 + c^5 = -5abc(ab + bc + ac)$$

$$a^6 + b^6 + c^6 = 3(abc)^2 - 2(ab + bc + ac)^3$$

$$a^7 + b^7 + c^7 = 7abc(ab + bc + ac)^2$$

$$\frac{a^5 + b^5 + c^5}{5} = (\frac{a^2 + b^2 + c^2}{2})(\frac{a^3 + b^3 + c^3}{3})$$

$$a^6 + b^6 + c^6 = 3(\frac{a^3 + b^3 + c^3}{3})^2 + 2(\frac{a^2 + b^2 + c^2}{2})$$

$$\frac{a^7 + b^7 + c^7}{7} = (\frac{a^2 + b^2 + c^2}{2})(\frac{a^5 + b^5 + c^5}{5})$$

Notable Implications

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

  $$x^2 + x^{-2} = a^2 - 2$$

  $$x^3 + x^{-3} = a^3 - 3a$$

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

Auxiliary Identities

$$a^3 + b^3 + c^3 = \frac{1}{2}(a + b + c)[(a - b)^2 + (a - c)^2 + (b - c)^2]$$

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

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

Practice Problems

Level I

Level II

Level III