Mathematical tables: short multiplication formulas
Tables shows various short multiplication formulas. Both common formulas such as (a + b)² (square of a sum) and general cases (e.g. any power of a sum) are presented.

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Short multiplication formulas#

NameFormula
Square of sumShow source(a+b)2=a2+2 ab+b2\left(a+\mathrm{b}\right)^{2}=a^{2}+2~a \cdot \mathrm{b}+\mathrm{b}^{2}
Square of differenceShow source(ab)2=a22 ab+b2\left(a-\mathrm{b}\right)^{2}=a^{2}-2~a \cdot \mathrm{b}+\mathrm{b}^{2}
Cube of a sumShow source(a+b)3=a3+3 a2b+3 ab2+b3\left(a+\mathrm{b}\right)^{3}=a^{3}+3~a^{2} \cdot \mathrm{b}+3~a \cdot \mathrm{b}^{2}+\mathrm{b}^{3}
Cube of a differenceShow source(ab)3=a33 a2b+3 ab2b3\left(a-\mathrm{b}\right)^{3}=a^{3}-3~a^{2} \cdot \mathrm{b}+3~a \cdot \mathrm{b}^{2}-\mathrm{b}^{3}
Sum of a cubesShow sourcea3+b3=(a+b)(a2ab+b2)a^{3}+\mathrm{b}^{3}=\left(a+\mathrm{b}\right) \cdot \left(a^{2}-a \cdot \mathrm{b}+\mathrm{b}^{2}\right)
Difference of a cubsShow sourcea3b3=(ab)(a2+ab+b2)a^{3}-\mathrm{b}^{3}=\left(a-\mathrm{b}\right) \cdot \left(a^{2}+a \cdot \mathrm{b}+\mathrm{b}^{2}\right)
Sophie Germain identityShow sourcea4+4 b4=(a2+2 ab+2 b2)(a22 ab+2 b2)a^{4}+4~\mathrm{b}^{4}=\left(a^{2}+2~a \cdot \mathrm{b}+2~\mathrm{b}^{2}\right) \cdot \left(a^{2}-2~a \cdot \mathrm{b}+2~\mathrm{b}^{2}\right)
Difference of a fourth powersShow sourcea4b4=(ab)(a3+a2b+ab2+b3)a^{4}-\mathrm{b}^{4}=\left(a-\mathrm{b}\right) \cdot \left(a^{3}+a^{2} \cdot \mathrm{b}+a \cdot \mathrm{b}^{2}+\mathrm{b}^{3}\right)
Sum of a fifth powersShow sourcea5+b5=(a+b)(a4a3b+a2b2ab3+b4)a^{5}+\mathrm{b}^{5}=\left(a+\mathrm{b}\right) \cdot \left(a^{4}-a^{3} \cdot \mathrm{b}+a^{2} \cdot \mathrm{b}^{2}-a \cdot \mathrm{b}^{3}+\mathrm{b}^{4}\right)
Difference of a fifth powersShow sourcea5b5=(ab)(a4+a3b+a2b2+ab3+b4)a^{5}-\mathrm{b}^{5}=\left(a-\mathrm{b}\right) \cdot \left(a^{4}+a^{3} \cdot \mathrm{b}+a^{2} \cdot \mathrm{b}^{2}+a \cdot \mathrm{b}^{3}+\mathrm{b}^{4}\right)
Square of three terms: (a + b + c)²Show source(a+b+c)2=a2+b2+c2+2 ab+2 ac+2 bc\left(a+\mathrm{b}+c\right)^{2}=a^{2}+\mathrm{b}^{2}+c^{2}+2~a \cdot \mathrm{b}+2~a \cdot c+2~\mathrm{b} \cdot c
Square of three terms: (a + b - c)²Show source(a+bc)2=a2+b2+c2+2 ab2 ac2 bc\left(a+\mathrm{b}-c\right)^{2}=a^{2}+\mathrm{b}^{2}+c^{2}+2~a \cdot \mathrm{b}-2~a \cdot c-2~\mathrm{b} \cdot c
Square of three terms: (a - b + c)²Show source(ab+c)2=a2+b2+c22 ab+2 ac2 bc\left(a-\mathrm{b}+c\right)^{2}=a^{2}+\mathrm{b}^{2}+c^{2}-2~a \cdot \mathrm{b}+2~a \cdot c-2~\mathrm{b} \cdot c
Square of three terms: (a - b - c)²Show source(ab+c)2=a2+b2+c22 ab2 ac+2 bc\left(a-\mathrm{b}+c\right)^{2}=a^{2}+\mathrm{b}^{2}+c^{2}-2~a \cdot \mathrm{b}-2~a \cdot c+2~\mathrm{b} \cdot c
Any power of a sum - general formulaShow source(a+b)n=k=0n(nk)ankbk=k=0nn!k!(nk)!ankbk(a + b)^n = \sum_{k=0}^{n}{n \choose k} a^{n-k} b^k = \sum_{k=0}^{n}{\frac{n!}{k!(n-k)!}} a^{n-k} b^k
Any power of a difference - general formulaShow source(ab)n=k=0n(1)k(nk)ankbk=k=0n(1)kn!k!(nk)!ankbk(a - b)^n = \sum_{k=0}^{n}{(-1)^k {n \choose k}} a^{n-k} b^k = \sum_{k=0}^{n}{(-1)^k \frac{n!}{k!(n-k)!}} a^{n-k} b^k
Square of a sum of any number of termsShow source(i=1kai)2=i=1kj=1kaiaj\left(\sum_{i=1}^{k}{a_i}\right)^2 = \sum_{i=1}^{k} \sum_{j=1}^{k} a_i a_j

Short multiplication formulas: formulas with square of a numbers#

NameFormula
Square of sumShow source(a+b)2=a2+2 ab+b2\left(a+\mathrm{b}\right)^{2}=a^{2}+2~a \cdot \mathrm{b}+\mathrm{b}^{2}
Square of differenceShow source(ab)2=a22 ab+b2\left(a-\mathrm{b}\right)^{2}=a^{2}-2~a \cdot \mathrm{b}+\mathrm{b}^{2}

Short multiplication formulas: formulas with cube of a numbers#

NameFormula
Cube of a sumShow source(a+b)3=a3+3 a2b+3 ab2+b3\left(a+\mathrm{b}\right)^{3}=a^{3}+3~a^{2} \cdot \mathrm{b}+3~a \cdot \mathrm{b}^{2}+\mathrm{b}^{3}
Cube of a differenceShow source(ab)3=a33 a2b+3 ab2b3\left(a-\mathrm{b}\right)^{3}=a^{3}-3~a^{2} \cdot \mathrm{b}+3~a \cdot \mathrm{b}^{2}-\mathrm{b}^{3}
Sum of a cubesShow sourcea3+b3=(a+b)(a2ab+b2)a^{3}+\mathrm{b}^{3}=\left(a+\mathrm{b}\right) \cdot \left(a^{2}-a \cdot \mathrm{b}+\mathrm{b}^{2}\right)
Difference of a cubsShow sourcea3b3=(ab)(a2+ab+b2)a^{3}-\mathrm{b}^{3}=\left(a-\mathrm{b}\right) \cdot \left(a^{2}+a \cdot \mathrm{b}+\mathrm{b}^{2}\right)

Short multiplication formulas: formulas with higher powers#

NameFormula
Sophie Germain identityShow sourcea4+4 b4=(a2+2 ab+2 b2)(a22 ab+2 b2)a^{4}+4~\mathrm{b}^{4}=\left(a^{2}+2~a \cdot \mathrm{b}+2~\mathrm{b}^{2}\right) \cdot \left(a^{2}-2~a \cdot \mathrm{b}+2~\mathrm{b}^{2}\right)
Difference of a fourth powersShow sourcea4b4=(ab)(a3+a2b+ab2+b3)a^{4}-\mathrm{b}^{4}=\left(a-\mathrm{b}\right) \cdot \left(a^{3}+a^{2} \cdot \mathrm{b}+a \cdot \mathrm{b}^{2}+\mathrm{b}^{3}\right)
Sum of a fifth powersShow sourcea5+b5=(a+b)(a4a3b+a2b2ab3+b4)a^{5}+\mathrm{b}^{5}=\left(a+\mathrm{b}\right) \cdot \left(a^{4}-a^{3} \cdot \mathrm{b}+a^{2} \cdot \mathrm{b}^{2}-a \cdot \mathrm{b}^{3}+\mathrm{b}^{4}\right)
Difference of a fifth powersShow sourcea5b5=(ab)(a4+a3b+a2b2+ab3+b4)a^{5}-\mathrm{b}^{5}=\left(a-\mathrm{b}\right) \cdot \left(a^{4}+a^{3} \cdot \mathrm{b}+a^{2} \cdot \mathrm{b}^{2}+a \cdot \mathrm{b}^{3}+\mathrm{b}^{4}\right)

Short multiplication formulas: three terms#

NameFormula
Square of three terms: (a + b + c)²Show source(a+b+c)2=a2+b2+c2+2 ab+2 ac+2 bc\left(a+\mathrm{b}+c\right)^{2}=a^{2}+\mathrm{b}^{2}+c^{2}+2~a \cdot \mathrm{b}+2~a \cdot c+2~\mathrm{b} \cdot c
Square of three terms: (a + b - c)²Show source(a+bc)2=a2+b2+c2+2 ab2 ac2 bc\left(a+\mathrm{b}-c\right)^{2}=a^{2}+\mathrm{b}^{2}+c^{2}+2~a \cdot \mathrm{b}-2~a \cdot c-2~\mathrm{b} \cdot c
Square of three terms: (a - b + c)²Show source(ab+c)2=a2+b2+c22 ab+2 ac2 bc\left(a-\mathrm{b}+c\right)^{2}=a^{2}+\mathrm{b}^{2}+c^{2}-2~a \cdot \mathrm{b}+2~a \cdot c-2~\mathrm{b} \cdot c
Square of three terms: (a - b - c)²Show source(ab+c)2=a2+b2+c22 ab2 ac+2 bc\left(a-\mathrm{b}+c\right)^{2}=a^{2}+\mathrm{b}^{2}+c^{2}-2~a \cdot \mathrm{b}-2~a \cdot c+2~\mathrm{b} \cdot c

Short multiplication formulas: general formulas#

NameFormula
Any power of a sum - general formulaShow source(a+b)n=k=0n(nk)ankbk=k=0nn!k!(nk)!ankbk(a + b)^n = \sum_{k=0}^{n}{n \choose k} a^{n-k} b^k = \sum_{k=0}^{n}{\frac{n!}{k!(n-k)!}} a^{n-k} b^k
Any power of a difference - general formulaShow source(ab)n=k=0n(1)k(nk)ankbk=k=0n(1)kn!k!(nk)!ankbk(a - b)^n = \sum_{k=0}^{n}{(-1)^k {n \choose k}} a^{n-k} b^k = \sum_{k=0}^{n}{(-1)^k \frac{n!}{k!(n-k)!}} a^{n-k} b^k
Square of a sum of any number of termsShow source(i=1kai)2=i=1kj=1kaiaj\left(\sum_{i=1}^{k}{a_i}\right)^2 = \sum_{i=1}^{k} \sum_{j=1}^{k} a_i a_j

Some facts#

  • The short multiplication formulas allow quick performing of common mathematical operations e.g. a square of the sum of two numbers:
    ⓘ Example: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
  • There is no obligation to use short multiplication formulas, because the same calculation can be done manually (step by step, multiplying all components one-by-one). However, the use of ready-made formulas may help to avoid tedious calculations and reduce the chance of making a mistake.
  • We can achieve the above formula by multiplying the terms one by one (colloquially: each by each):
    (a+b)2=(a+b)(a+b)=a(a+b)+b(a+b)=aa+ab+ba+bb=a2+2ab+b2 \begin{array}{l} (a + b)^2 = \\ (a + b)(a + b) = \\ a(a + b) + b(a + b) = \\ a \cdot a + ab + ba + b \cdot b = \\ a^2 + 2ab + b^2 \end{array}

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