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Binomial

For the scientific naming of living things, see binomial nomenclature.

See binomial (disambiguation) for a list of other meanings.

In elementary algebra, a binomial is a polynomial with two terms: the sum of two monomials. It is the simplest kind of polynomial.

Examples:

$a + b \quad$
$x+3 \quad$
${x \over 2} + {x^2 \over 2}$
$v t - {1 \over 2} g t^2$

The product of a binomial a + b with a factor c is obtained by distributing the monomial:

$c (a + b) = c a + c b \$

The product of two binomials a + b and c + d is obtained by distributing twice:

$(a + b)(c + d) = (a + b) c + (a + b) d \$

$= a c + b c + a d + b d \quad$ .

The square of a binomial a + b is

$(a + b)^2 = a^2 + 2 a b + b^2 \quad$

and the square of the binomial a - b is

$(a - b)^2 = a^2 - 2 a b + b^2. \quad$

The binomial $a 2 - b 2$ can be factored as the product of two other binomials:

$a^2 - b^2 = (a + b)(a - b). \quad$

A binomial is linear if it is of the form

$a x + b \quad$

where a and b are constants and x is a variable.

A complex number is a binomial of the form

$a + i b \quad$

where i is the square root of minus one.

The product of a pair of linear binomials a x + b and c x + d is:

$a x + b \quad$

$c x + d \quad$

$----------- \quad$

$a c x^2 + \ \ \ c b \, x \quad$

$\ \ \ \ \ a d x \ \ \ \ \ \, + b d \quad$

$----------- \quad$

$a c x^2 + (c b + a d) x + b d \quad$

A binomial a + b raised to the n^th power, represented as

$(a + b)^n \quad$

can be expanded by means of the binomial theorem or Pascal's triangle. Pascal's triangle is not good to use with large numbers but as a rule of thumb will suffice where the power does not exceed 7.

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Binomial

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