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p is an integer factor of the constant term a 0, and q is an integer factor of the leading coefficient a n . The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials.
For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x 2 y 2. However, a polynomial in variables x and y, is a polynomial in x with coefficients which are polynomials in y, and also a polynomial in y with coefficients which are polynomials in x. The polynomial
As another example, in radix 5, a string of digits such as 132 denotes the (decimal) number 1 × 5 2 + 3 × 5 1 + 2 × 5 0 = 42. This representation is unique. This representation is unique. Let b be a positive integer greater than 1.
Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent.
For k = 0 and k = n, Descartes' rule of signs shows that the polynomial has exactly one positive real root. If R 0 {\displaystyle R_{0}} and R n {\displaystyle R_{n}} are these roots, the above result shows that all the roots satisfy
One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
Jurors spent more than 13 hours deliberating their verdict before reaching a majority agreement, with 10 finding Vickers guilty of murder. He was remanded in custody to be sentenced on 10 February.
A constant coefficient, also known as constant term or simply constant, is a quantity either implicitly attached to the zeroth power of a variable or not attached to other variables in an expression; for example, the constant coefficients of the expressions above are the number 3 and the parameter c, involved in 3=c ⋅ x 0.