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Continued fractions are most conveniently applied to solve the general quadratic equation expressed in the form of a monic polynomial x 2 + b x + c = 0 {\displaystyle x^{2}+bx+c=0} which can always be obtained by dividing the original equation by its leading coefficient .
The result is an equation with no fractions. The simplified equation is not entirely equivalent to the original. For when we substitute y = 0 and z = 0 in the last equation, both sides simplify to 0, so we get 0 = 0, a mathematical truth.
Continued fractions can also be applied to problems in number theory, and are especially useful in the study of Diophantine equations. In the late eighteenth century Lagrange used continued fractions to construct the general solution of Pell's equation, thus answering a question that had fascinated mathematicians for more than a thousand years. [9]
The formula for the difference of two squares can be used for factoring polynomials that contain the square of a first quantity minus the square of a second quantity. For example, the polynomial can be factored as follows:
where x is a variable we are interested in solving for, we can use cross-multiplication to determine that x = b c d . {\displaystyle x={\frac {bc}{d}}.} For example, suppose we want to know how far a car will travel in 7 hours, if we know that its speed is constant and that it already travelled 90 miles in the last 3 hours.
A simple fraction (also known as a common fraction or vulgar fraction) [n 1] is a rational number written as a/b or , where a and b are both integers. [9] As with other fractions, the denominator (b) cannot be zero. Examples include 1 / 2 , − 8 / 5 , −8 / 5 , and 8 / −5 .
Using this approach, solving a polynomial of degree is related to the ways of rearranging ("permuting") terms, called the symmetric group on letters and denoted . For the quadratic polynomial, the only ways to rearrange two roots are to either leave them be or to transpose them, so solving a quadratic polynomial ...
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
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