Special Products

Perfect Square Trinomial

Perfect Square Trinomial – equationzone.com

(a+b)2=a2+2ab+b2\boxed{(a + b)^2 = a^2 + 2ab + b^2}

(a−b)2=a2−2ab+b2\boxed{(a - b)^2 = a^2 - 2ab + b^2}

Important:

(a−b)2=(b−a)2(a - b)^2 = (b - a)^2

  • (x+3)2=x2+2(x)(3)+32=x2+6x+9(x + 3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9
  • (2a−5)2=(2a)2−2(2a)(5)+52=4a2−20a+25(2a - 5)^2 = (2a)^2 - 2(2a)(5) + 5^2 = 4a^2 - 20a + 25
  • (y+1)2=y2+2(y)(1)+12=y2+2y+1(y + 1)^2 = y^2 + 2(y)(1) + 1^2 = y^2 + 2y + 1
  • (3x−2y)2=(3x)2−2(3x)(2y)+(2y)2=9x2−12xy+4y2(3x - 2y)^2 = (3x)^2 - 2(3x)(2y) + (2y)^2 = 9x^2 - 12xy + 4y^2
  • (a+12)2=a2+2(a)(12)+(12)2=a2+a+14(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=52−2(5)(b)+b2=25−10b+b2(5 - b)^2 = 5^2 - 2(5)(b) + b^2 = 25 - 10b + b^2

Legendre's Identities

(a+b)2+(a−b)2=2(a2+b2)(a+b)^2+(a-b)^2=2(a^2+b^2)

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

(a+b)4−(a−b)4=8ab(a2+b2)(a + b)^4 - (a - b)^4 = 8ab(a^2 + b^2)

Difference of Squares

Difference of Squares – equationzone.com

a2−b2=(a+b)(a−b)\boxed{a^2 - b^2 = (a + b)(a - b)}

  • x2−9=x2−32=(x+3)(x−3)x^2 - 9 = x^2 - 3^2 = (x + 3)(x - 3)
  • 4a2−25=(2a)2−52=(2a+5)(2a−5)4a^2 - 25 = (2a)^2 - 5^2 = (2a + 5)(2a - 5)
  • 16y2−1=(4y)2−12=(4y+1)(4y−1)16y^2 - 1 = (4y)^2 - 1^2 = (4y + 1)(4y - 1)
  • 49−z2=72−z2=(7+z)(7−z)49 - z^2 = 7^2 - z^2 = (7 + z)(7 - z)
  • x24−y29=(x2)2−(y3)2=(x2+y3)(x2−y3)\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)2−42=(x+2+4)(x+2−4)=(x+6)(x−2)(x + 2)^2 - 16 = (x + 2)^2 - 4^2 = (x + 2 + 4)(x + 2 - 4) = (x + 6)(x - 2)

Square of a Trinomial

Square of a Trinomial – equationzone.com

(a+b+c)2=a2+b2+c2+2ab+2ac+2bc\boxed{(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc}

(a+b+c)2=a2+b2+c2+2(ab+ac+bc)\boxed{(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc)}

  • (x+y+2)2=x2+y2+4+2xy+4x+4y(x + y + 2)^2 = x^2 + y^2 + 4 + 2xy + 4x + 4y
  • (a+3+b)2=a2+9+b2+6a+2ab+6b(a + 3 + b)^2 = a^2 + 9 + b^2 + 6a + 2ab + 6b
  • (1+m+n)2=1+m2+n2+2m+2n+2mn(1 + m + n)^2 = 1 + m^2 + n^2 + 2m + 2n + 2mn
  • (2a+3b+1)2=4a2+9b2+1+12ab+4a+6b(2a + 3b + 1)^2 = 4a^2 + 9b^2 + 1 + 12ab + 4a + 6b
  • (x−1+y)2=x2+1+y2−2x+2xy−2y(x - 1 + y)^2 = x^2 + 1 + y^2 - 2x + 2xy - 2y
  • (p+q+r)2=p2+q2+r2+2pq+2pr+2qr(p + q + r)^2 = p^2 + q^2 + r^2 + 2pq + 2pr + 2qr

Other Forms

(a−b+c)2=a2+b2+c2−2ab+2ac−2bc(a - b + c)^2 = a^2 + b^2 + c^2 - 2ab + 2ac - 2bc

(a+b−c)2=a2+b2+c2+2ab−2ac−2bc(a + b - c)^2 = a^2 + b^2 + c^2 + 2ab - 2ac - 2bc

(a−b−c)2=a2+b2+c2−2ab−2ac+2bc(a - b - c)^2 = a^2 + b^2 + c^2 - 2ab - 2ac + 2bc

(a−b−c)2=(−(b+c−a))2=(b+c−a)2(a - b - c)^2 = ( -(b + c - a) )^2 = (b + c - a)^2

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

Cube of a Binomial

(a+b)3=a3+3a2b+3ab2+b3\boxed{(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3}

(a−b)3=a3−3a2b+3ab2−b3\boxed{(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3}

  • (x+2)3=x3+3x2(2)+3x(4)+8=x3+6x2+12x+8(x + 2)^3 = x^3 + 3x^2(2) + 3x(4) + 8 = x^3 + 6x^2 + 12x + 8
  • (a−1)3=a3−3a2(1)+3a(1)−1=a3−3a2+3a−1(a - 1)^3 = a^3 - 3a^2(1) + 3a(1) - 1 = a^3 - 3a^2 + 3a - 1
  • (2x+y)3=(2x)3+3(4x2)(y)+3(2x)(y2)+y3=8x3+12x2y+6xy2+y3(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=a3−3a2(3b)+3a(9b2)−27b3=a3−9a2b+27ab2−27b3(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+3x2+x3(1 + x)^3 = 1 + 3x + 3x^2 + x^3
  • (x−y)3=x3−3x2y+3xy2−y3(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3

Compact Forms (Cauchy’s Identities)

(a+b)3=a3+b3+3ab(a+b)(a + b)^3 = a^3 + b^3 + 3ab(a + b)

(a−b)3=a3−b3−3ab(a−b)(a - b)^3 = a^3 - b^3 - 3ab(a - b)

(a+b)3+(a−b)3=2a(a2+3b2)(a + b)^3 + (a - b)^3 = 2a(a^2 + 3b^2)

(a+b)3−(a−b)3=2b(3a2+b2)(a + b)^3 - (a - b)^3 = 2b(3a^2 + b^2)

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

(x+a)(x+b)=x2+(a+b)x+ab\boxed{(x + a)(x + b) = x^2 + (a + b)x + ab}

(x−a)(x−b)=x2−(a+b)x+ab(x - a)(x - b) = x^2 - (a + b)x + ab

(x+a)(x+b)(x+c)=x3+(a+b+c)x2+(ab+bc+ac)x+abc(x + a)(x + b)(x + c) = x^3 + (a + b + c)x^2 + (ab + bc + ac)x + abc

(x−a)(x−b)(x−c)=x3−(a+b+c)x2+(ab+bc+ac)x−abc(x - a)(x - b)(x - c) = x^3 - (a + b + c)x^2 + (ab + bc + ac)x - abc

  • (x+3)(x+5)=x2+(3+5)x+3⋅5=x2+8x+15(x + 3)(x + 5) = x^2 + (3 + 5)x + 3 \cdot 5 = x^2 + 8x + 15
  • (x+2)(x−7)=x2+(2−7)x+2⋅(−7)=x2−5x−14(x + 2)(x - 7) = x^2 + (2 - 7)x + 2 \cdot (-7) = x^2 - 5x - 14
  • (x−4)(x−6)=x2+(−4−6)x+(−4)⋅(−6)=x2−10x+24(x - 4)(x - 6) = x^2 + (-4 - 6)x + (-4) \cdot (-6) = x^2 - 10x + 24
  • (x+1)(x+9)=x2+(1+9)x+1⋅9=x2+10x+9(x + 1)(x + 9) = x^2 + (1 + 9)x + 1 \cdot 9 = x^2 + 10x + 9
  • (x−3)(x+8)=x2+(−3+8)x+(−3)⋅8=x2+5x−24(x - 3)(x + 8) = x^2 + (-3 + 8)x + (-3) \cdot 8 = x^2 + 5x - 24
  • (x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^2 + (a + b)x + ab

Cube of a Trinomial

(a+b+c)3=a3+b3+c3+3(a+b)(b+c)(c+a)\boxed{(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)}

Alternative Forms:

(a+b+c)3=a3+b3+c3+3(a+b)(b+c)(c+a)−3abc(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a) - 3abc

(a+b+c)3=3(a+b+c)(a2+b2+c2)−2(a3+b3+c3)+6abc(a + b + c)^3 = 3(a+b+c)(a^2+b^2+c^2)-2(a^3+b^3+c^3)+6abc

(a+b+c)3=a3+b3+c3+3a2(b+c)+3b2(a+c)+3c2(a+b)+6abc(a + b + c)^3 = a^3 + b^3 + c^3 + 3a^2(b + c) + 3b^2(a + c) + 3c^2(a + b) + 6abc

  • (x+1+2)3=x3+13+23+3(x+1)(1+2)(2+x)=x3+9+9(x+1)(x+2)(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=13+23+33+3(1+2)(2+3)(3+1)=36+180=216(1 + 2 + 3)^3 = 1^3 + 2^3 + 3^3 + 3(1 + 2)(2 + 3)(3 + 1) = 36 + 180 = 216
  • (a+0+b)3=a3+b3+3ab(a+b)=a3+3a2b+3ab2+b3(a + 0 + b)^3 = a^3 + b^3 + 3ab(a + b) = a^3 + 3a^2b + 3ab^2 + b^3
  • (x+y+0)3=x3+y3+3xy(x+y)(x + y + 0)^3 = x^3 + y^3 + 3xy(x + y)
  • (1+x+1)3=(x+2)3=x3+8+6(x+1)2(1 + x + 1)^3 = (x + 2)^3 = x^3 + 8 + 6(x + 1)^2
  • (a+b+c)3=a3+b3+c3+3(a+b)(b+c)(c+a)(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a)

Sum and Difference of Cubes

a3+b3=(a+b)(a2−ab+b2)\boxed{a^3 + b^3 = (a + b)(a^2 - ab + b^2)}

a3−b3=(a−b)(a2+ab+b2)\boxed{a^3 - b^3 = (a - b)(a^2 + ab + b^2)}

  • 8+27=23+33=(2+3)(4−6+9)=5⋅7=358 + 27 = 2^3 + 3^3 = (2 + 3)(4 - 6 + 9) = 5 \cdot 7 = 35
  • x3−8=x3−23=(x−2)(x2+2x+4)x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)
  • 64+125=43+53=(4+5)(16−20+25)=9⋅21=18964 + 125 = 4^3 + 5^3 = (4 + 5)(16 - 20 + 25) = 9 \cdot 21 = 189
  • 27a3−64b3=(3a)3−(4b)3=(3a−4b)(9a2+12ab+16b2)27a^3 - 64b^3 = (3a)^3 - (4b)^3 = (3a - 4b)(9a^2 + 12ab + 16b^2)
  • 1+x3=(1+x)(1−x+x2)1 + x^3 = (1 + x)(1 - x + x^2)
  • y3+1000=y3+103=(y+10)(y2−10y+100)y^3 + 1000 = y^3 + 10^3 = (y + 10)(y^2 - 10y + 100)

Argand’s Identities

(a2+a+1)(a2−a+1)=a4+a2+1\boxed{(a^2 + a + 1)(a^2 - a + 1) = a^4 + a^2 + 1}

(a2+ab+b2)(a2−ab+b2)=a4+a2b2+b4\boxed{(a^2 + ab + b^2)(a^2 - ab + b^2) = a^4 + a^2b^2 + b^4}

In General:

(a2m+ambn+b2n)(a2m−ambn+b2n)=a4m+a2mb2n+b4n(a^{2m} + a^m b^n + b^{2n})(a^{2m} - a^m b^n + b^{2n}) = a^{4m} + a^{2m}b^{2n} + b^{4n}

Gauss’s Identity

a3+b3+c3−3abc=(a+b+c)(a2+b2+c2−ab−ac−bc)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 + b)(b + c)(a + c) + abc = (a + b + c)(ab + ac + bc)

a2+b2+c2−ab−ac−bc=12[(a−b)2+(a−c)2+(b−c)2]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=(a2+b2)(x2+y2)(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=(a2+b2+c2)(x2+y2+z2)(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 an+bna^n + b^n

a2+b2=(a+b)2−2aba^2 + b^2 = (a + b)^2 - 2ab

a3+b3=(a+b)3−3ab(a+b)a^3 + b^3 = (a + b)^3 - 3ab(a + b)

a4+b4=(a+b)4−4ab(a+b)2+2(ab)2a^4 + b^4 = (a + b)^4 - 4ab(a + b)^2 + 2(ab)^2

a5+b5=(a+b)5−5ab(a+b)3+5(ab)2(a+b)a^5 + b^5 = (a + b)^5 - 5ab(a + b)^3 + 5(ab)^2(a + b)

Conditional Identities

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

a2+b2+c2=−2(ab+ac+bc)a^2 + b^2 + c^2 = -2(ab + ac + bc)

a3+b3+c3=3abca^3 + b^3 + c^3 = 3abc

(ab+ac+bc)2=a2b2+a2c2+b2c2(ab + ac + bc)^2 = a^2b^2 + a^2c^2 + b^2c^2

(a2+b2+c2)2=2(a4+b4+c4)(a^2 + b^2 + c^2)^2 = 2(a^4 + b^4 + c^4)

a4+b4+c4=2(a2b2+a2c2+b2c2)=12(a2+b2+c2)2a^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

a5+b5+c5=−5abc(ab+bc+ac)a^5 + b^5 + c^5 = -5abc(ab + bc + ac)

a6+b6+c6=3(abc)2−2(ab+bc+ac)3a^6 + b^6 + c^6 = 3(abc)^2 - 2(ab + bc + ac)^3

a7+b7+c7=7abc(ab+bc+ac)2a^7 + b^7 + c^7 = 7abc(ab + bc + ac)^2

a5+b5+c55=(a2+b2+c22)(a3+b3+c33)\frac{a^5 + b^5 + c^5}{5} = (\frac{a^2 + b^2 + c^2}{2})(\frac{a^3 + b^3 + c^3}{3})

a6+b6+c6=3(a3+b3+c33)2+2(a2+b2+c22)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})

a7+b7+c77=(a2+b2+c22)(a5+b5+c55)\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 ab+ba=2\dfrac{a}{b} + \dfrac{b}{a} = 2, then:

    a=ba = b

  • If a2+b2+c2=ab+ac+bca^2 + b^2 + c^2 = ab + ac + bc, then:

    a=b=ca = b = c

  • If a3+b3+c3=3abca^3 + b^3 + c^3 = 3abc and a+b+c=0a + b + c = 0, then:

    a=b=c=0a = b = c = 0

  • If a2+b2+c2+⋯+z2=0a^2 + b^2 + c^2 + \dots + z^2 = 0, then:

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

  • If an+bn+⋯+zn=0\sqrt[n]{a} + \sqrt[n]{b} + \dots + \sqrt[n]{z} = 0, then:

    a=b=⋯=z=0a = b = \dots = z = 0

  • If x+x−1=ax + x^{-1} = a, then:

    x2+x−2=a2−2x^2 + x^{-2} = a^2 - 2

    x3+x−3=a3−3ax^3 + x^{-3} = a^3 - 3a

    x4+x−4=(a2−2)2−2x^4 + x^{-4} = (a^2 - 2)^2 - 2

Auxiliary Identities

a3+b3+c3=12(a+b+c)[(a−b)2+(a−c)2+(b−c)2]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−a3−b3−c3=3(a+b)(b+c)(c+a)(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=(a2+b2+c2)−2(ab+ac+bc)(a - b)^2 + (b - c)^2 + (a - c)^2 = (a^2 + b^2 + c^2) - 2(ab + ac + bc)