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For example, the coefficient of x 5 is +1 because there are two ways to split 5 into an even number of distinct parts (4 + 1 and 3 + 2), but only one way to do so for an odd number of distinct parts (the one-part partition 5). However, the coefficient of x 12 is −1 because there are seven ways to partition 12 into an even number of distinct ...
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
In mathematics, LHS is informal shorthand for the left-hand side of an equation.Similarly, RHS is the right-hand side.The two sides have the same value, expressed differently, since equality is symmetric.
In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem.
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. [1] This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!"
A cyclic permutation consisting of a single 8-cycle. There is not widespread consensus about the precise definition of a cyclic permutation. Some authors define a permutation σ of a set X to be cyclic if "successive application would take each object of the permuted set successively through the positions of all the other objects", [1] or, equivalently, if its representation in cycle notation ...