Trinomial expansion

Last updated October 15, 2024

In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by

Derivation

The trinomial expansion can be calculated by applying the binomial expansion twice, setting $d=b+c$ , which leads to

{\begin{aligned}(a+b+c)^{n}&=(a+d)^{n}=\sum _{r=0}^{n}{n \choose r}\,a^{n-r}\,d^{r}\\&=\sum _{r=0}^{n}{n \choose r}\,a^{n-r}\,(b+c)^{r}\\&=\sum _{r=0}^{n}{n \choose r}\,a^{n-r}\,\sum _{s=0}^{r}{r \choose s}\,b^{r-s}\,c^{s}.\end{aligned}}

Above, the resulting $(b+c)^{r}$ in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index $s$ .

The product of the two binomial coefficients is simplified by shortening $r!$ ,

{n \choose r}\,{r \choose s}={\frac {n!}{r!(n-r)!}}{\frac {r!}{s!(r-s)!}}={\frac {n!}{(n-r)!(r-s)!s!}},

and comparing the index combinations here with the ones in the exponents, they can be relabelled to $i=n-r,~j=r-s,~k=s$ , which provides the expression given in the first paragraph.

Properties

The number of terms of an expanded trinomial is the triangular number

t_{n+1}={\frac {(n+2)(n+1)}{2}},

where $n$ is the exponent to which the trinomial is raised.^[3]

Example

An example of a trinomial expansion with $n=2$ is :

$(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ca$

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References

↑ Koshy, Thomas (2004), Discrete Mathematics with Applications, Academic Press, p. 889, ISBN 9780080477343 .
↑ Harris, John; Hirst, Jeffry L.; Mossinghoff, Michael (2009), Combinatorics and Graph Theory, Undergraduate Texts in Mathematics (2nd ed.), Springer, p. 146, ISBN 9780387797113 .
↑ Rosenthal, E. R. (1961), "A Pascal pyramid for trinomial coefficients", The Mathematics Teacher, 54 (5): 336–338, doi:10.5951/MT.54.5.0336 .

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[1] Koshy, Thomas (2004), Discrete Mathematics with Applications, Academic Press, p. 889, ISBN 9780080477343 .

[2] Harris, John; Hirst, Jeffry L.; Mossinghoff, Michael (2009), Combinatorics and Graph Theory, Undergraduate Texts in Mathematics (2nd ed.), Springer, p. 146, ISBN 9780387797113 .

[3] Rosenthal, E. R. (1961), "A Pascal pyramid for trinomial coefficients", The Mathematics Teacher, 54 (5): 336–338, doi:10.5951/MT.54.5.0336 .

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