Binomial

Notable products between binomials

Notable products are used in algebra for the literal calculation of the product between binomials. They are said to be notable because the product of some particular polynomials always reaches the same result. For this reason, it is possible to avoid, for these particular polynomials, the carrying out of all the calculation steps of the product, and therefore write directly the result as established by the remarkable products.

Multiplying a binomial by itself

The result of the product between two similar binomials in which one of the two presents the sum operator, while the other the subtraction operator, is a combination consisting of the difference between the square of the first monomial and the square of the second:

\[(a+b)(a-b)=a^{2}-b^{2}\]

It is possible to find this remarkable product case, even with powers, but the calculation process does not change:

\[(a+b)^{3}(a-b)^{3}=(a^{2}-b^{2})^{3}\]

The power remains unchanged and is enclosed in parentheses in the notable product developed. Then the binomial cube is applied and the expression is resolved as follows:

\[(a^{2}-b^{2})^{3}=a^{6}-3a^{4}b^{2}+3a^{2}b^{4}-b^{6}\]

Square of a binomial

The square of a binomial (square of the sum and square of the subtraction) turns out to be a trinomial having as its terms the square of the first term, the double product of the first term for the second and the square of the second:

\[(a+b)^{2}=a^{2}+2ab+b^{2}\]

\[(a-b)^{2}=a^{2}-2ab+b^{2}\]

Cube of a binomial

The cube of a binomial (cube of the sum and the cube of the subtraction) turns out to be a quadrinomium having as its terms the cube of the first term, the triple of the square of the first term for the second, the triple of the first term for the square of the second, the cube of the second term:

\[(a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}\]

\[(a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}\]

\[a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})\]

\[a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})\]

Nth power of a binomial (binomial theorem)

Two tools are used to calculate the nth power of a binomial; the first is the formula below (also called binomial theorem), and the second is Pascal’s triangle. The Binomial Theorem is a quick way of expanding (or multiplying out) a binomial expression that has been raised to some (generally inconveniently large) power.

Pascal’s triangle (PDF)Download

\[(a+b)^{n}=\sum_{k=0}^{n}a^{(n-k)}b^{k}\]

Demonstration for \(n=4\):

\[\begin{align*}
(a+b)^{4} & =\sum_{k=0}^{4}\binom{4}{k}a^{4-k}b^{k}=\binom{4}{0}a^{4-0}b^{0}+\\
& +\binom{4}{1}a^{4-1}b^{1}+\binom{4}{2}a^{4-2}b^{2}+\\
& +\binom{4}{3}a^{4-3}b^{3}+\binom{4}{4}a^{4-4}b^{4}=\\
& =a^{4}+a^{3}b+a^{2}b^{2}+ab^{3}+b^{4}
\end{align*}\]

References

  1. Pascal’s triangle and Sierpinski triangle. TeXample. http://texample.net/tikz/examples/pascals-triangle-and-sierpinski-triangle/

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