The formula for abbreviated multiplication

The formula for abbreviated multiplication

The most important abbreviated multiplication formulas.

Formulas of abbreviated multiplication allow you to perform calculations much faster.
The most commonly used abbreviated multiplication formulas:

(a + b)2 = a2 + 2from + b2

(a − b)2 = a2 2from + b2

(a+b+c)2 = a2 + b2 + c2 + 2from + 2ac + 2bc

a2 b2 = (a + b)(a b)

(a + b)3 = a3 + 3a2b + 3from2 + b3

(a b)3 = a3 3a2b + 3from2 b3

a3 + b3 = (a + b)(a2 from + b2)

a3 b3 = (ab)(a2 + from + b2)

 

The abbreviated multiplication formulas are helpful for multiplying or expanding algebraic expressions. They facilitate efficient counting. There are a lot of these patterns. We list a few below, which are used most often.

The square of the sum of the numbers

  • (a + b)2 = a2 + 2from + b2
    e.g: 312 = (30+1)2 = 302+2×30+1 = 900+60+1 = 961
  • does not occur equality: (a+b)2 = a2 + b2
    e.g 25 = (3+2)2 32 + 22 = 13
  • justification of the formula by the bill:
    (a + b)2 = (a + b) × (a + b) =
    aa + from + ba + bb = a2 + 2from + b2

The square of the difference of numbers

  • (a b)2 = a2 – 2from + b2
    e.g: 292 = (30-1)2 = 302-2×30+1 = 900-60+1 = 841
  • does not occur equality: (a-b)2 = a2b2
    e.g 1 = (3-2)2 32 – 22 = 5
  • justification of the formula:
    (a – b)2 = (a – b) × (a – b) = aa from ba + bb = a2 – 2from + b2

     

A square of the sum of three numbers

  • (a+b+c)2 = a2 + b2 + c2 + 2from + 2ac + 2bc
    e.g: 1112 = (100+10+1)2 = 1002 + 102 +1 +2×100×10 + 2×100 + 2×10 = 10000 + 100 + 1 + 2000 + 200 + 20 = 12321
  • does not occur equality: (a+b+c)2 = a2 + b2 + c2
    e.g 36 = (3+2+1)2 32 + 22 + 12 = 14
  • justification of the formula:
    (a + b + c)2 = (a + b + c) × (a + b + c) = aa + from + ac + ba + bb + bc + that + cb + cc = a2 + b2 + c2 + 2from + 2ac + 2bc

Product of the sum and difference of numbers = Difference of squares of numbers

  • (a + b)×(ab) = a2 b2
    e.g: 101×99 = (100+1)×(100-1) = 1002 – 1 = 9999
  • justification of the formula :
    (a + b) × (a – b) = aa from + ba bb = a2 b2

A cube of the sum of numbers

  • (a + b)3 = a3 + 3a2b + 3from2 + b3
    e.g: 1013 = (100+1)3 = 1003 + 3×1002 + 3×100 + 1 =
    = 1000000 + 30000 + 300 + 1 = 1030301
  • does not occur equality: (a+b)3 = a3 + b3
    e.g 125 = (3+2)3 33 + 23 = 35
  • justification of the formula by the bill:
    (a + b)3 = (a + b) × (a + b) × (a + b)
    = (aa + from + ba + bb) × (a + b) = aaa + aab + aba + fig + baa + chapter + bba + bbb =
    = a3 + 3a2b + 3from2 + b3

Cube of number difference

  • (a b)3 = a3 – 3a2b + 3from2 b3
    e.g: 993 = (100-1)3 = 1003 – 3×1002 + 3×100 – 1 =
    = 1000000 – 30000 + 300 – 1 = 970299

Sum of cubes of numbers

a3 + b3 = (a + b)×(a2 from + b2)

justification of the formula:

(a + b)×(a2 from + b2) = aa2 aab + from2 + ba2 chapter + bb2= a3a2b + from2 + a2bfrom2 + b3 =
= a3 + b3

The difference of the cubes of numbers

a3 b3 = (ab)×(a2 + from + b2)

justification of the formula:

(ab)×(a2 + from + b2) = aa2 + aab + from2 ba2 chapterbb2 = a3 + a2b + from2 a2bfrom2 b3 =
= a3b3

Difference of fourth powers of numbers

a4 b4 = (ab)×(a3 + a2b + from2 + b3) = (a + b)×(a3a2b + from2b3)

 

Sum n-these powers of numbers (for n odd!!!)

an + bn = (a + b) (an-1an-2b + an-3b2 – … + bn-1)

 

Difference n-these powers of numbers (for n even!!!)

anbn = (a + b) (an-1an-2b + an-3b2 – … + bn-1)

 

Difference n-these powers of numbers (for everyone n natural)

anbn = (a b) (an-1 + an-2b + an-3b2 + … + a2bn-3 + fromn-2 + bn-1)