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The leading-order terms (or corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude. [ 1 ] [ 2 ] The sizes of the different terms in the equation(s) will change as the variables change, and hence, which terms are leading-order may also change.
Divide the highest term of the remainder by the highest term of the divisor (3x ÷ x = 3). Place the result (+3) below the bar. 3x has been divided leaving no remainder, and can therefore be marked as used. The result 3 is then multiplied by the second term in the divisor −3 = −9. Determine the partial remainder by subtracting −4 − (− ...
The leading term of a polynomial is thus the term of the largest monomial (for the chosen monomial ordering). Concretely, let R be any ring of polynomials. Then the set M of the (monic) monomials in R is a basis of R , considered as a vector space over the field of the coefficients.
The leading entry (that is, the left-most nonzero entry) of every nonzero row, called the pivot, is on the right of the leading entry of every row above. [ 2 ] Some texts add the condition that the leading coefficient must be 1 [ 3 ] while others require this only in reduced row echelon form .
The leading term of e λ t (X 1, ..., X n) is X λ. Proof. The leading term of the product is the product of the leading terms of each factor (this is true whenever one uses a monomial order, like the lexicographic order used here), and the leading term of the factor e i (X 1, ..., X n) is clearly X 1 X 2 ···X i.
So if two leading coefficients are in the same column, then a row operation of type 3 could be used to make one of those coefficients zero. Then by using the row swapping operation, one can always order the rows so that for every non-zero row, the leading coefficient is to the right of the leading coefficient of the row above.
For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x ...
If f(n) = n 2 + 3n, then as n becomes very large, the term 3n becomes insignificant compared to n 2. The function f(n) is said to be "asymptotically equivalent to n 2, as n → ∞". This is often written symbolically as f (n) ~ n 2, which is read as "f(n) is asymptotic to n 2". An example of an important asymptotic result is the prime number ...