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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.
In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials [1] —hence the method may be referred to as the FOIL method. The word FOIL is an acronym for the four terms of the product: First ("first" terms of each binomial are multiplied together)
When a monomial order has been chosen, the leading monomial is the largest u in S, the leading coefficient is the corresponding c u, and the leading term is the corresponding c u u. Head monomial/coefficient/term is sometimes used as a synonym of "leading". Some authors use "monomial" instead of "term" and "power product" instead of "monomial".
A primitive monomial is a special case of a monomial in this second sense, where the coefficient is . For example, in this interpretation − 7 x 5 {\displaystyle -7x^{5}} and ( 3 − 4 i ) x 4 y z 13 {\displaystyle (3-4i)x^{4}yz^{13}} are monomials (in the second example, the variables are x , y , z , {\displaystyle x,y,z,} and the coefficient ...
Using the multiplication tables embedded in the rods, multiplication can be reduced to addition operations and division to subtractions. Advanced use of the rods can extract square roots. Napier's bones are not the same as logarithms, with which Napier's name is also associated, but are based on dissected multiplication tables.
In binary (base-2) math, multiplication by a power of 2 is merely a register shift operation. Thus, multiplying by 2 is calculated in base-2 by an arithmetic shift. The factor (2 −1) is a right arithmetic shift, a (0) results in no operation (since 2 0 = 1 is the multiplicative identity element), and a (2 1) results in a left arithmetic shift ...
For example, to multiply 7 and 15 modulo 17 in Montgomery form, again with R = 100, compute the product of 3 and 4 to get 12 as above. The extended Euclidean algorithm implies that 8⋅100 − 47⋅17 = 1, so R′ = 8. Multiply 12 by 8 to get 96 and reduce modulo 17 to get 11. This is the Montgomery form of 3, as expected.
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.