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In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials [1] ... (or fewer, if like terms are then combined) monomials. [6]
A monomial, also called a power product or primitive monomial, [1] ... the set of monomials is a subset of all polynomials that is closed under multiplication.
As with the monomials, one would set up the sides of the rectangle to be the factors and then fill in the rectangle with the algebra tiles. [2] This method of using algebra tiles to multiply polynomials is known as the area model [3] and it can also be applied to multiplying monomials and binomials with each other.
In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., If and is any other monomial, then .
The multiplication of a polynomial by a scalar consists of multiplying each coefficient by this scalar, without any other change in the representation. The multiplication of a polynomial by a monomial m consists of multiplying each monomial of the polynomial by m. This does not change the term ordering by definition of a monomial ordering.
It is a polynomial in which no variable occurs to a power of or higher; that is, each monomial is a constant times a product of distinct variables. For example f ( x , y , z ) = 3 x y + 2.5 y − 7 z {\displaystyle f(x,y,z)=3xy+2.5y-7z} is a multilinear polynomial of degree 2 {\displaystyle 2} (because of the monomial 3 x y {\displaystyle 3xy ...
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This implies that, the monic polynomials in a univariate polynomial ring over a commutative ring form a monoid under polynomial multiplication. Two monic polynomials are associated if and only if they are equal, since the multiplication of a polynomial by a nonzero constant produces a polynomial with this constant as its leading coefficient.