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Suppose that y 0 = 1, y 1, ... is a sequence of polynomials where y n has degree n. If this is a sequence of orthogonal polynomials for some positive weight function then it satisfies a 3-term recurrence relation. Favard's theorem is roughly a converse of this, and states that if these polynomials satisfy a 3-term recurrence relation of the form
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation.Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. [1]
A similar problem, involving equating like terms rather than coefficients of like terms, arises if we wish to de-nest the nested radicals + to obtain an equivalent expression not involving a square root of an expression itself involving a square root, we can postulate the existence of rational parameters d, e such that
The polynomial 3x 2 − 5x + 4 is written in descending powers of x. The first term has coefficient 3, indeterminate x, and exponent 2. In the second term, the coefficient is −5. The third term is a constant. Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two. [11]
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...
All the above multiplication algorithms can also be expanded to multiply polynomials. Alternatively the Kronecker substitution technique may be used to convert the problem of multiplying polynomials into a single binary multiplication. [31] Long multiplication methods can be generalised to allow the multiplication of algebraic formulae:
The third power of the trinomial a + b + c is given by (+ +) = + + + + + + + + +. This can be computed by hand using the distributive property of multiplication over addition and combining like terms, but it can also be done (perhaps more easily) with the multinomial theorem.
In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors.For example, 21 is the product of 3 and 7 (the result of multiplication), and (+) is the product of and (+) (indicating that the two factors should be multiplied together).