A team led by a mathematician from the University of New South Wales (UNSW) in Sydney has discovered a new method to tackle one of the oldest challenges in algebra: solving higher-degree polynomial equations.<br /><br />In algebra, polynomial equations involve variables raised to powers (e.g., x² or x³). Simple second-degree equations (like x²) have been solved since around 1800 BCE by the Babylonians, who developed the technique known as "completing the square."<br /><br />Later, in his book Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala, Al-Khwarizmi laid out the first clear rules for solving such equations. These methods eventually evolved into the quadratic formula familiar to many high school students today.<br /><br />In the 16th century, mathematicians developed general methods to solve cubic and quartic equations. However, higher-degree equations—those of degree five or above (e.g., x⁵ or higher)—have resisted general algebraic solutions.<br /><br />These equations are fundamental in mathematics and science, with applications in describing planetary motion, computer programming, and more. Therefore, solving higher-degree polynomials would bring significant benefits to these fields.<br /><br />Norman Wildberger and Higher-Degree Polynomials<br />In 1832, French mathematician Évariste Galois proved that general exact solutions to fifth-degree (and higher) polynomial equations are impossible due to their mathematical complexity. Since then, approximate solutions have been developed and widely used in applications, but these do not fall under the realm of pure algebra.<br /><br />However, mathematician Norman Wildberger and his team at UNSW in Sydney have discovered a novel approach to solving high-degree equations. Their findings were published in the journal The American Mathematical Monthly.<br /><br />Wildberger noted that traditional methods rely on roots (like square roots and cube roots), which often produce irrational numbers—numbers that can’t be calculated exactly because they contain an infinite number of decimal places.<br /><br />Instead of relying on roots, Wildberger’s method uses power series—equations that contain an infinite number of terms involving powers of "x." Power series are a clever way to represent any mathematical function as a sum of smaller, increasingly fine components.<br /><br />To visualize it, imagine building a tower using blocks: you place one block, then a smaller one, and then an even smaller one, and so on. Each block adds a bit more height. Similarly, in a power series, the final answer is built step-by-step by adding smaller and smaller numerical contributions.<br /><br />Wildberger then truncated these series to extract approximate numerical answers and tested the method on historically known equations. It worked remarkably well—including on a famous cubic equation from the 17th century used in Newton’s method.<br /><br />Catalan Numbers and Combinatorics<br />The new method also employs sequences of numbers that represent complex geometric relationships. These sequences come from combinatorics, a branch of mathematics that deals with counting patterns in sets of elements.<br /><br />One of the most famous combinatorial sequences is the Catalan numbers, which count how many ways a polygon (a shape with three or more sides) can be divided into triangles.<br /><br />Catalan numbers are special values that help determine the number of valid ways to arrange or divide objects without errors. In other words, they are useful for solving certain types of structured problems.<br /><br />For example, imagine dividing a pizza-shaped polygon with multiple sides: the question becomes, "In how many ways can this pizza be sliced into triangles without any intersecting lines?"<br /><br />If the pizza has 3 sides, there’s only 1 way.<br /><br />With 4 sides, there are 2 ways (using either diagonal).<br /><br />With 5 sides, there are 5 ways.<br /><br />And so on.<br /><br />These counts are the Catalan numbers.<br /><br /><br />"AL_mustaqbal University is the first university in Iraq"<br/><br/><a href=https://uomus.edu.iq/Default.aspx target=_blank>al-mustaqbal University Website</a>